Math module

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i
Mathematics
Learner’s Material
Unit 1
Department of Education
Republic of the Philippines
9
This instructional material was collaboratively
developed and reviewed by educators from public and
private schools, colleges, and/or universities. We
encourage teachers and other education stakeholders
to email their feedback, comments, and
recommendations to the Department of Education at
action@deped.gov.ph.
We value your feedback and recommendations.

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Mathematics – Grade 9
Learner’s Material
First Edition, 2014
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Published by the Department of Education
Secretary: Br. Armin A. Luistro FSC
Undersecretary: Dina S. Ocampo, Ph.D.
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E-mail Address: imcsetd@yahoo.com

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TABLE OF CONTENTS
QUARTER 1
Module No. 1: Quadratic Equations and Inequalities
Lesson 1 Illustrations of Quadratic Equations 11
Lesson 2A Solving Quadratic Equations by
Extracting Square Roots
19
Lesson 2B Solving Quadratic Equations by
Factoring
30
Lesson 2C Solving Quadratic Equations by
Completing The Square
39
Lesson 2D Solving Quadratic Equations By Using
The Quadratic Formula
53
Lesson 3 The Nature of the Roots of a Quadratic
Equation
65
Lesson 4 The Sum and the Product of Roots of
Quadratic Equations
77
Lesson 5 Equations Transformable to Quadratic
Equations
90
Lesson 6 Solving Problems Involving Quadratic
Equations
104
Lesson 7 Quadratic Inequalities 114
Module 2. Quadratic Functions
Lesson 1 Introduction to Quadratic Function 146
Lesson 2 Graphs of Quadratic Functions 164
Lesson 3 Finding the Equation of a Quadratic
Function
181
Lesson 4 Applications of Quadratic Functions 200

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LEARNING MODULE
MATH 9
MODULE NO. 1: QUADRATIC EQUATIONS AND
INEQUALITIES
INTRODUCTION AND FOCUS QUESTIONS:
Was there any point in your life when you asked yourself about the different
real life quantities such as costs of goods or services, incomes, profits, yields and
losses, amount of particular things, speed, area, and many others? Have you ever
realized that these quantities can be mathematically represented to come up with
practical decisions?
Find out the answers to these questions and determine the vast applications
of quadratic equations and quadratic inequalities through this module.
LESSONS AND COVERAGE:
In this module, you will examine the above questions when you take the following
lessons:
Lesson 1 – ILLUSTRATIONS OF QUADRATIC EQUATIONS
Lesson 2 – SOLVING QUADRATIC EQUATIONS
 EXTRACTING SQUARE ROOTS
 FACTORING
 COMPLETING THE SQUARE
 QUADRATIC FORMULA
SALES
PROFITS

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Lesson 3 – NATURE OF ROOTS OF QUADRATIC EQUATIONS
Lesson 4 – SUM AND PRODUCT OF ROOTS OF QUADRATIC EQUATIONS
Lesson 5 – EQUATIONS TRANSFORMABLE TO QUADRATIC EQUATIONS
(INCLUDING RATIONAL ALGEBRAIC EQUATIONS)
Lesson 6 – APPLICATIONS OF QUADRATIC EQUATIONS AND RATIONAL
ALGEBRAIC EQUATIONS
Lesson 7 – QUADRATIC INEQUALITIES
In these lessons, you will learn to:
Lesson 1  illustrate quadratic equations;
Lesson 2  solve quadratic equations by: (a) extracting square roots; (b)
factoring; (c) completing the square; (d) using the quadratic
formula;
Lesson 3  characterize the roots of a quadratic equation using the
discriminant;
Lesson 4  describe the relationship between the coefficients and the roots
of a quadratic equation;
Lesson 5  solve equations transformable to quadratic equations (including
rational algebraic equations);
Lesson 6  solve problems involving quadratic equations and rational
algebraic equations;
Lesson 7  illustrate quadratic inequalities;
 solve quadratic inequalities; and
 solve problems involving quadratic inequalities.

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MODULE MAP:
Here is a simple map of the lessons that will be covered in this module:
Quadratic Equations, Quadratic Inequalities,
and Rational Algebraic Equations
Illustrations of
Quadratic Equations
Solving Quadratic
Equations
Extracting Square Roots
Factoring
Completing the Square
Quadratic Formula
Nature of Roots of
Quadratic Equations
Sum and Product of
Roots of Quadratic
Equations
Equations
Transformable to
Quadratic Equations
Applications of
Quadratic Equations
and Rational
Algebraic Equations
Illustrations of
Quadratic Inequalities
Solving Quadratic
Inequalities
Applications of
Rational Algebraic Equations

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PRE-ASSESSMENT
Part I
Directions: Find out how much you already know about this module. Choose the
letter that you think best answers the question. Please answer all items. Take note of
the items that you were not able to answer correctly and find the right answer as you
go through this module.
1. It is a polynomial equation of degree two that can be written in the form
ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
A. Linear Equation C. Quadratic Equation
B. Linear Inequality D. Quadratic Inequality
2. Which of the following is a quadratic equation?
A. 142 2
 rr C. 01452
 ss
B. 1273 t D. 372 2
 xx
3. In the quadratic equation 0473 2
 xx , which is the quadratic term?
A. 2
x B. x7 C. 2
3x D. 4
4. Which of the following rational algebraic equations is transformable to a quadratic
equation?
A. 7
4
2
2
1



 ww
C.
4
3
2
1
3
12 qq


B. 5
1
32



pp
D.
mmm
7
2
4
2
3




5. How many real roots does the quadratic equation 0752
 xx have?
A. 0 B. 1 C. 2 D. 3
6. The roots of a quadratic equation are -5 and 3. Which of the following quadratic
equations has these roots?
A. 01582
 xx C. 01522
 xx
B. 01582
 xx D. 01522
 xx
7. Which of the following mathematical statements is a quadratic inequality?
A. 0532 2
 rr C. 0273 2
 tt
B. 0127 h D. 01582
 ss

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8. Which of the following shows the graph of 672
 xxy ?
A. C.
B. D.
9. Which of the following values of x make the equation 01872
 xx true?
I. -9 II. 2 III. 9
A. I and II B. II and III C. I and III D. I, II, and III
10. Which of the following quadratic equations has no real roots?
A. 342 2
 xx C. 523 2
 ss
B. 0482
 tt D. 072 2
 rr
11.What is the nature of the roots of the quadratic equation if the value of its
discriminant is zero?
A. The roots are not real. C. The roots are rational and not
equal.
B. The roots are irrational and not equal. D. The roots are rational and equal.
y
x
y
x
y
x
y
x

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12.One of the roots of 020132 2
 xx is 4. What is the other root?
A.
5
2
 B.
2
5
 C.
5
2
D.
2
5
13.What are the roots of the quadratic equation 0562
 xx ?
A. 2 and -1 B. 8 and -7 C. -8 and 7 D. 3 and -2
14.What is the sum of the roots of the quadratic equation 01462 2
 xx ?
A. -7 B. -3 C. 6 D. 14
15.Which of the following quadratic equations can be solved easily by extracting
square roots?
A. 01272
 xx C. 094 2
t
B. 0372 2
 ww D. 0823 2
 vv
16.Which of the following coordinates of points belong to the solution set of the
inequality 152 2
 xxy ?
A. (-3,2) B. (-2,9) C. (1,6) D. (3,1)
17.A 3 cm by 3 cm square piece of cardboard was cut from a bigger square
cardboard. The area of the remaining cardboard was 40 cm2. If s represents the
length of the bigger cardboard, which of the following expressions give the area
of the remaining piece?
A. 9s B. 92
s C. 92
s D. 402
s
18.The length of a wall is 12 m more than its width. If the area of the wall is less than
50 m2, which of the following could be its length?
A. 3 m B. 4 m C. 15 m D. 16 m
19.The length of a garden is 5 m longer than its width and the area is 14 m2. How
long is the garden?
A. 9 m B. 7 m C. 5 m D. 2 m
20.A car travels 20 kph faster than a truck. The car covers 480 km in two hours less
than the time it takes the truck to travel the same distance. How fast does the car
travel?
A. 44 kph B. 60 kph C. 80 kph D. 140 kph
21.A 12 cm by 16 cm picture is mounted with border of uniform width on a
rectangular frame. If the total area of the border is 288 cm2, what is the length of
the side of the frame?
A. 8 cm B. 16 cm C. 20 cm D. 24 cm

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22.SamSon Electronics Company would like to come up with an LED TV such that
its screen is 560 square inches larger than the present ones. Suppose the length
of the screen of the larger TV is 6 inches longer than its width and the area of the
smaller TV is 520 square inches. What is the length of the screen of the larger
LED TV?
A. 26 in. B. 30 in C. 33 in. D. 36 in.
23.
A. I and II B. I and III C. II and III D. I, II, and III
24.It takes Mary 3 hours more to do a job than it takes Jane. If they work together,
they can finish the same job in 2 hours. How long would it take Mary to finish the
job alone?
A. 3 hours B. 5 hours C. 6 hours D. 8 hours
25.An open box is to be formed out of a rectangular piece of cardboard whose
length is 12 cm longer than its width. To form the box, a square of side 5 cm will
be removed from each corner of the cardboard. Then the edges of the remaining
cardboard will be turned up. If the box is to hold at most 1900 cm3, what
mathematical statement would represent the given situation?
A. 360122
 xx C. 40082
 xx
B. 380122
 xx D. 40082
 xx
26.The length of a garden is 2 m more than twice its width and its area is 24 m2.
Which of the following equations represents the given situation?
A. 122
 xx C. 242
 xx
B. 1222
 xx D. 2422
 xx
The figure on the right shows the graph of
142 2
 xxy . Which of the following is
true about the solution set of the inequality?
I. The coordinates of all points on the
shaded region belong to the solution
set of the inequality.
II. The coordinates of all points along the
parabola as shown by the broken line
belong to the solution set of the
inequality.
III. The coordinates of all points along the
parabola as shown by the broken line
do not belong to the solution set of the
inequality.
y
x

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27.From 2004 through 2012, the average weekly income of an employee in a certain
company is estimated by the quadratic expression 224044.516.0 2
 nn , where
n is the number of years after 2004. In what year was the average weekly income
of an employee equal to Php2,271.20?
A. 2007 B. 2008 C. 2009 D. 2010
28.In the figure below, the area of the shaded region is 144 cm2. What is the length
of the longer side of the figure?
A. 8 cm B. 12 cm C. 14 cm D. 18 cm
6 cm
4 cm
s
s

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Part II
Directions: Read and understand the situation below then answer or perform what
are asked.
Mrs. Villareal was asked by her principal to transfer her Grade 9 class to a
new classroom that was recently built. The room however still does not have fixtures
such as teacher’s table, bulletin boards, divan, bookshelves, and cabinets. To help
her acquire these fixtures, she informed the parents of her students about these
classroom needs. The parents decided to donate construction materials such as
wood, plywood, nails, paints, and many others.
After all the materials have been received, she asked her students to make
the designs of the different classroom needs. Each group of students was assigned
to do the design of a particular fixture. The designs that the students will prepare
shall be used by the carpenter in constructing the tables, chairs, bulletin boards,
divan, bookshelves, and cabinets.
1. Suppose you are one of the students of Mrs. Villareal. How will you prepare the
design of one of the fixtures?
2. Make a design of the fixture assigned to your group.
3. Illustrate every part or portion of the fixture including their measurements.
4. Using the design of the fixture made, determine all the mathematics concepts or
principles involved.
5. Formulate problems involving these mathematics concepts or principles.
6. Write the expressions, equations, or inequalities that describe the situations or
problems.
7. Solve the equations, the inequalities, and the problems formulated.
Rubric for Design
4 3 2 1
The design is
accurately made,
presentable, and
appropriate.
The design is
accurately made
and appropriate.
The design is not
accurately made
but appropriate.
The design is
made but not
appropriate.

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Rubric for Equations Formulated and Solved
4 3 2 1
Equations and
inequalities are
properly formulated
and solved
correctly.
Equations and
inequalities are
properly formulated
but not all are
solved correctly.
Equations and
inequalities are
properly formulated
but are not solved
correctly.
Equations and
inequalities are
properly
formulated but
are not solved.
Rubric on Problems Formulated and Solved
Score Descriptors
6
Poses a more complex problem with 2 or more correct possible
solutions and communicates ideas unmistakably, shows in-depth
comprehension of the pertinent concepts and/or processes and
provides explanations wherever appropriate.
5
Poses a more complex problem and finishes all significant parts of
the solution and communicates ideas unmistakably, shows in-
depth comprehension of the pertinent concepts and/or processes.
4
Poses a complex problem and finishes all significant parts of the
solution and communicates ideas unmistakably, shows in-depth
comprehension of the pertinent concepts and/or processes.
3
Poses a complex problem and finishes most significant parts of the
solution and communicates ideas unmistakably, shows
comprehension of major concepts although neglects or
misinterprets less significant ideas or details.
2
Poses a problem and finishes some significant parts of the solution
and communicates ideas unmistakably but shows gaps on
theoretical comprehension.
1
Poses a problem but demonstrates minor comprehension, not
being able to develop an approach.
Source: D.O. #73 s. 2012

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LEARNING GOALS AND TARGETS:
After going through this module, you should be able to demonstrate
understanding of key concepts of quadratic equations, quadratic inequalities, and
rational algebraic equations, formulate real-life problems involving these concepts,
and solve these using a variety of strategies. Furthermore, you should be able to
investigate mathematical relationships in various situations involving quadratic
equations and quadratic inequalities.
LESSON NO. 1: ILLUSTRATIONS OF QUADRATIC EQUATIONS
What to KNOW:
Activity 1: Do You Remember These Products?
Directions: Find each indicated product then answer the questions that follow.
1.  73 2
x 6.   44  xx
2.  42 ss 7.   5252  rr
3.   37  ww 8.  2
43 m
4.   29  xx 9.   7272  hh
5.   512  tt 10.   xx 3838 
Questions:
a. How did you find each product?
b. In finding each product, what mathematics concepts or principles did you apply?
Explain how you applied these mathematics concepts or principles.
c. How would you describe the products obtained?
Start Lesson 1 of this module by assessing your knowledge of the different
mathematics concepts previously studied and your skills in performing
mathematical operations. These knowledge and skills will help you in
understanding quadratic equations. As you go through this lesson, think of this
important question: “How are quadratic equations used in solving real-life
problems and in making decisions?” To find the answer, perform each activity. If
you find any difficulty in answering the exercises, seek the assistance of your
teacher or peers or refer to the modules you have gone over earlier. You may
check your work with your teacher.

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Are the products polynomials? If YES, what common characteristics do these
polynomials have?
Activity 2: Another Kind of Equation!
Directions: Below are different equations. Use these equations to answer the
questions that follow.
1. Which of the given equations are linear?
2. How do you describe linear equations?
3. Which of the given equations are not linear? Why?
How are these equations different from those which are linear?
What common characteristics do these equations have?
Activity 3: A Real Step to Quadratic Equations
Directions: Use the situation below to answer the questions that follow.
Mrs. Jacinto asked a carpenter to construct a rectangular bulletin board for
her classroom. She told the carpenter that the board’s area must be 18 square feet.
1. Draw a diagram to illustrate the bulletin board.
2. What are the possible dimensions of the bulletin board? Give at least 2 pairs of
possible dimensions.
3. How did you determine the possible dimensions of the bulletin board?
Were you able to find and describe the products of some polynomials? Were you
able to recall and apply the different mathematics concepts or principles in
finding each product? Why do you think there is a need to perform such
mathematical tasks? You will find this out as you go through this lesson.
0352
 xx
732  ts
0259 2
r
83
2
1 2
 xx
512  nc
106  qp
1549  x
06
4
3
h
1238 k 0144 2
 mm 0672
 tt 1442
r
In the activity you have just done, were you able to identify equations which are
linear and which are not? Were you able to describe those equations which are
not linear? These equations have common characteristics and you will learn
more of these in the succeeding activities.

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4. Suppose the length of the board is 7 ft. longer than its width. What equation
would represent the given situation?
5. How would you describe the equation formulated?
6. Do you think you can use the equation formulated to find the length and the width
of the bulletin board? Justify your answer.
In the equation, ax2 is the quadratic term, bx is the linear term, and c is the
constant term.
Example 1: 2x2 + 5x – 3 = 0 is a quadratic equation in standard form with a =
2, b = 5, and c = -3.
Example 2: 3x(x – 2) = 10 is a quadratic equation. However, it is not written
in standard form.
To write the equation in standard form, expand the product and
make one side of the equation zero as shown below.
  1023 xx  1063 2
 xx
10101063 2
 xx
01063 2
 xx
The equation becomes 3x2 – 6x – 10 = 0, which is in standard
form.
In the equation 3x2 – 6x – 10 = 0, a = 3, b = -6, and c = -10.
How did you find the preceding activities? Are you ready to learn about quadratic
equations? I’m sure you are!!! From the activities done, you were able to
describe equations other than linear equations, and these are quadratic
equations. You were able to find out how a particular quadratic equation is
illustrated in real life. But how are quadratic equations used in solving real-life
problems and in making decisions? You will find these out in the activities in the
next section. Before doing these activities, read and understand first some
important notes on quadratic equations and the examples presented.
A quadratic equation in one variable is a
mathematical sentence of degree 2 that can be
written in the following standard form.
ax2 + bx + c = 0, where a, b, and c are real
numbers and a ≠ 0
Why do you think a
must not be equal to
zero in the equation
ax2 + bx + c = 0?

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14
Example 3: The equation (2x + 5)(x – 1) = -6 is also a quadratic equation but
is not written in standard form.
Just like in Example 2, the equation (2x + 5)(x – 1) = -6 can be
written in standard form by expanding the product and making
one side of the equation zero as shown below.
   6152  xx  65522 2
 xxx
6532 2
 xx
666532 2
 xx
0132 2
 xx
The equation becomes 2x2 + 3x + 1 = 0, which is in standard
form.
In the equation 2x2 + 3x + 1 = 0, a = 2, b = 3, and c = 1.
When b = 0 in the equation ax2 + bx + c = 0, it results to a quadratic equation
of the form ax2 + c = 0.
Examples: Equations such as x2 + 5 = 0, -2x2 + 7 = 0, and 16x2 – 9 = 0 are
quadratic equations of the form ax2 + c = 0. In each equation,
the value of b = 0.
What to PROCESS:
Your goal in this section is to apply the key concepts of quadratic equations. Use
the mathematical ideas and the examples presented in the preceding section to
answer the activities provided.
Learn more about
Quadratic Equations
through the WEB. You
may open the following
links.
http://library.thinkquest.org/20991/alg2/quad
.html
http://math.tutorvista.com/algebra/quadratic-
equation.html
http://www.algebra.com/algebra/homework/
quadratic/lessons/quadform/

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15
Activity 4: Quadratic or Not Quadratic?
Directions: Identify which of the following equations are quadratic and which are not.
If the equation is not quadratic, explain.
1. 1583 m 6. rr 425 2

2. 01052
 xx 7.   723 xx
3. 0412  x 8.   06
2
1
h
4. 1272 2
 tt 9.   02 2
x
5. 0326 2
 xx 10.    1458  ww
Activity 5: Does It Illustrate Me?
Directions: Tell whether or not each of the following situations illustrates quadratic
equations. Justify your answer by representing each situation by a mathematical
sentence.
1. The length of a swimming pool is 8 m longer than its width and the area is 105
m2.
2. Edna paid at least Php1,200 for a pair of pants and a blouse. The cost of the pair
of pants is Php600 more than the cost of the blouse.
3. A motorcycle driver travels 15 kph faster than a bicycle rider. The motorcycle
driver covers 60 km in two hours less than the time it takes the bicycle rider to
travel the same distance.
4. A realty developer sells residential lots for Php4,000 per square meter plus a
processing fee of Php25,000. One of the lots the realty developer is selling costs
Php625,000.
5. A garden 7 m by 12 m will be expanded by planting a border of flowers. The
border will be of the same width around the entire garden and has an area of 92
m2.
Were you able to identify which equations are quadratic? Some of the equations
given are not quadratic equations. Were you able to explain why? I’m sure you
did. In the next activity, you will identify the situations that illustrate quadratic
equations and represent these by mathematical statements.
Did you find the activity challenging? Were you able to represent each situation
by a mathematical statement? For sure you were able to identify the situations
that can be represented by quadratic equations. In the next activity, you will write
quadratic equations in standard form.

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16
Activity 6: Set Me To Your Standard!
Directions: Write each quadratic equation in standard form, ax2 + bx + c = 0 then
identify the values of a, b, and c. Answer the questions that follow.
1. 723 2
 xx 6.    xxx 377 
2. xx 625 2
 7.   084 2
x
3.    043  xx 8.    232 2
 xx
4.    0172  xx 9.    22
112  xx
5.   1532 xx 10.     3342  xxxx
Questions:
a. How did you write each quadratic equation in standard form?
b. What mathematics concepts or principles did you apply to write each quadratic
equation in standard form? Discuss how you applied these mathematics
concepts or principles.
c. Which quadratic equations did you find difficult to write in standard form? Why?
d. Compare your work with those of your classmates. Did you arrive at the same
answers? If NOT, explain.
How was the activity you have just done? Was it easy for you to write quadratic
equations in standard form? It was easy for sure!
In this section, the discussion was about quadratic equations, their forms and
how they are illustrated in real life.
Go back to the previous section and compare your initial ideas with the
discussion. How much of your initial ideas are found in the discussion? Which
ideas are different and need revision?
Now that you know the important ideas about this topic, let’s go deeper by
moving on to the next section.

DRAFT
March 24, 2014
17
What to REFLECT ON and FURTHER UNDERSTAND:
Activity 7: Dig Deeper!
Directions: Answer the following questions.
1. How are quadratic equations different from linear equations?
2. How do you write quadratic equations in standard form? Give at least 3
examples.
3. The following are the values of a, b, and c that Edna and Luisa got when they
expressed 2
235 xx  in standard form.
Edna: a = 2; b = 3; c = -5
Luisa: a = -2; b = -3; c = 5
Who do you think got the correct values of a, b, and c? Justify your answer.
4. Do you agree that the equation 2
234 xx  can be written in standard form in two
different ways? Justify your answer.
5.
a. How are you going to represent the number of Mathematics Club members?
b. What expression represents the amount each member will share?
c. If there were 25 members more in the club, what expression would represent
the amount each would share?
d. What mathematical sentence would represent the given situation? Write this
in standard form then describe.
Your goal in this section is to take a closer look at some aspects of the topic. You
are going to think deeper and test further your understanding of quadratic
equations. After doing the following activities, you should be able to answer this
important question: How are quadratic equations used in solving real-life
problems and in making decisions?
The members of the school’s Mathematics
Club shared equal amounts for a new
Digital Light Processing (DLP) projector
amounting to Php25,000. If there had
been 25 members more in the club, each
would have contributed Php50 less.

DRAFT
March 24, 2014
18
What to TRANSFER:
Activity 8: Where in the Real World?
1. Give 5 examples of quadratic equations written in standard form. Identify the
values of a, b, and c in each equation.
2. Name some objects or cite situations in real life where quadratic equations are
illustrated. Formulate quadratic equations out of these objects or situations then
describe each.
Summary/Synthesis/Generalization:
This lesson was about quadratic equations and how they are illustrated in real
life. The lesson provided you with opportunities to describe quadratic equations using
practical situations and their mathematical representations. Moreover, you were given
the chance to formulate quadratic equations as illustrated in some real-life situations.
Your understanding of this lesson and other previously learned mathematics concepts
and principles will facilitate your learning of the next lesson, Solving Quadratic
Equations.
Your goal in this section is to apply your learning to real-life situations. You will
be given a practical task which will demonstrate your understanding of quadratic
equations.
In this section, the discussion was about your understanding of quadratic
equations and how they are illustrated in real life.
What new realizations do you have about quadratic equations? How would you
connect this to real life? How would you use this in making decisions?
In this section, your task was to give examples of quadratic equations written in
standard form and name some objects or cite real-life situations where quadratic
equations are illustrated.
How did you find the performance task? How did the task help you realize the
importance of the topic in real life?
Now that you have a deeper understanding of the topic, you are ready to do the
tasks in the next section.

DRAFT
March 24, 2014
19
LEARNING MODULE
MATH 9
LESSON NO. 2A: SOLVING QUADRATIC EQUATIONS BY EXTRACTING
SQUARE ROOTS
What to KNOW:
Activity 1: Find My Roots!!!
Directions: Find the following square roots. Answer the questions that follow.
1. 16 = 6. 289 =
2. 25 = 7. 16.0 =
3. 49 = 8. 36 =
4. 64 = 9.
25
16
=
5. 121= 10.
256
169
 =
Questions:
a. How did you find each square root?
b. How many square roots does a number have? Explain your answer.
c. Does a negative number have a square root? Why?
d. Describe the following numbers: 8 , 40 , 60 , and 90
Start Lesson 2A of this module by assessing your knowledge of the different
mathematical operations. These knowledge and skills will help you in solving
quadratic equations by extracting square roots. As you go through this lesson,
think of this important question: “How does finding solutions of quadratic
equations facilitate in solving real-life problems and in making decisions?” To
find the answer, perform each activity. If you find any difficulty in answering the
exercises, seek the assistance of your teacher or peers or refer to the modules
you have gone over earlier. You may check your answers with your teacher.

DRAFT
March 24, 2014
20
Are the numbers rational or irrational? Explain your answer.
How do you describe rational numbers? How about numbers that are irrational?
Activity 2: What Would Make A Statement True?
Directions: Solve each of the following equations in as many ways as you can.
Answer
the questions that follow.
1. 127 x 6. 355  x
2. 104 t 7. 1623 h
3. 35 r 8. 287  x
4. 210 x 9.   2473 x
5. 162 s 10.   28132 k
Questions:
a. How did you solve each equation?
b. What mathematics concepts or principles did you apply to come up with the
solution of each equation? Explain how you applied these.
c. Compare the solutions you got with those of your classmates. Did you arrive at
the same answers? If not, why?
d. Which equations did you find difficult to solve? Why?
Were you able to find the square roots of some numbers? Did the activity provide
you with an opportunity to strengthen your understanding of rational and
irrational numbers? In the next activity, you will be solving linear equations. Just
like finding square roots of numbers, solving linear equations is also a skill which
you need to develop further in order for you understand the new lesson.
How did you find the activity? Were you able to recall and apply the different
mathematics concepts or principles in solving linear equations? I’m sure you
were. In the next activity, you will be representing a situation using a
mathematical sentence. Such mathematical sentence will be used to satisfy the
conditions of the given situation.

DRAFT
March 24, 2014
21
Activity 3: Air Out!!!
Directions: Use the situation below to answer the
1. Draw a diagram to illustrate the given situation.
2. How are you going to represent the length of a side of the square-shaped wall?
How about its area?
3. Suppose the area of the remaining part of the wall after the carpenter has made
the square opening is 6 m2. What equation would describe the area of the
remaining part of the wall?
4. How will you find the length of a side of the wall?
Activity 4: Learn to Solve Quadratic Equations!!!
Directions: Use the quadratic equations below to answer the questions that follow.
1. Describe and compare the given equations. What statements can you make?
2. Solve each equation in as many ways as you can. Determine the values of each
variable to make each equation true.
x2
= 36 t2
– 64 = 0 2s2
– 98 = 0
Mr. Cayetano plans to install a new exhaust
fan on his room’s square-shaped wall. He
asked a carpenter to make a square opening
on the wall where the exhaust fan will be
installed. The square opening must have an
area of 0.25 m2.
The activity you have just done shows how a real-life situation can be
represented by a mathematical sentence. Were you able to represent the given
situation by an equation? Do you now have an idea on how to use the equation
in finding the length of a side of the wall? To further give you ideas in solving the
equation or other similar equations, perform the next activity.

DRAFT
March 24, 2014
22
3. How did you know that the values of the variable really satisfy the equation?
4. Aside from the procedures that you followed in solving each equation, do you
think there are other ways of solving it? Describe these ways if there are any.
Activity 5: Anything Real or Nothing Real?
Directions: Find the solutions of each of the following quadratic equations, then
answer the questions that follow.
1. How did you determine the solutions of each equation?
2. How many solutions does each equation have? Explain your answer.
3. What can you say about each quadratic equation based on the solutions
obtained?
Quadratic equations that can be written in the form x2 = k can be solved by
applying the following properties:
How did you find the preceding activities? Are you ready to learn about solving
quadratic equations by extracting square roots? I’m sure you are!!! From the
activities done, you were able to find the square roots of numbers, solve linear
equations, represent a real-life situation by a mathematical sentence, and use
different ways of solving a quadratic equation. But how does finding solutions of
quadratic equations facilitate in solving real-life problems and in making
decisions? You will find these out in the activities in the next section. Before
doing these activities, read and understand first some important notes on
solving quadratic equations by extracting square roots and the examples
presented.
Were you able to determine the values of the variable that make each equation
true? Were you able to find other ways of solving each equation? Let us extend
your understanding of quadratic equations and learn more about their solutions
by performing the next activity.
x2
= 9 r2
= 0 w2
= -9
1. If k > 0, then kx 2
has two real solutions or roots: kx  .
2. If k = 0, then kx 2
has one real solution or root: 0x .
3. If k < 0, then kx 2
has no real solutions or roots.

DRAFT
March 24, 2014
23
The method of solving the quadratic equation kx 2
is called extracting
square roots.
Example 1: Find the solutions of the equation x2 – 16 = 0 by extracting
square roots.
Write the equation in the form kx 2
.
x2 – 16 = 0  x2 – 16 + 16 = 0 + 16
x2 = 16
Since 16 is greater than 0, then the first property above can be
applied to find the values of x that will make the equation
x2 – 16 = 0 true.
x2 = 16  16x
x = ±4
To check, substitute these values in the original equation.
Both values of x satisfy the given equation. So the equation
x2 – 16 = 0 is true when x = 4 or when x = -4.
Answer: The equation x2 – 16 = 0 has two solutions: x = 4 or x = -4.
Example 2: Solve the equation t2 = 0.
Since t2 equals 0, then the equation has only one solution.
That is, t = 0.
To check: 02
t
00
?
2

00 
Answer: The equation t2 = 0 has one solution: t = 0.
For x = 4:
0162
x
0164
?
2

01616
?

00 
For x = -4:
0162
x
  0164
?
2

01616
?

00 

DRAFT
March 24, 2014
24
Example 3: Find the roots of the equation 092
s .
Write the equation in the form kx 2
.
092
s  90992
s
92
s
Since -9 is less than 0, then the equation 92
s has no real
solutions or roots. There is no real number when squared gives -
9.
Answer: The equation 092
s has no real solutions or roots.
Example 4: Find the solutions of the equation (x – 4)2 – 25 = 0.
To solve (x – 4)2 – 25 = 0, add 25 on both sides of the equation.
(x – 4)2 – 25 + 25 = 0 + 25
The resulting equation is (x – 4)2 = 25.
Solve the resulting equation.
(x – 4)2 = 25  254 x
x – 4 = ±5
Solve for x in the equation x – 4 = ±5.
x – 4 + 4 = ±5 + 4  x = ±5 + 4
The equation will result to two values of x.
Check the obtained values of x against the original equation.
(x – 4)2 – 25 = 0 is true when x = 9 or when x = -1.
Answer: The equation (x – 4)2 – 25 = 0 has two solutions: x = 9 or x = -1
x = 5 + 4
x = 9
x = -5 + 4
x = -1
For x = 9:
  0254 2
x
  02549
?
2

0255
?
2

02525
?

00 
For x = -1:
  0254 2
x
  02541
?
2

  0255
?
2

02525
?

00 

DRAFT
March 24, 2014
25
What to PROCESS:
Activity 6: Extract Me!!!
Directions: Solve the following quadratic equations by extracting square roots.
Answer
the questions that follow.
1. x2 = 16 6. 4x2 – 225 = 0
2. t2 = 81 7. 3h2 – 147 = 0
3. r2 – 100 = 0 8. (x – 4)2 = 169
4. x2 – 144 = 0 9. (k + 7)2 = 289
5. 2s2 = 50 10. (2s – 1)2 = 225
Questions:
a. How did you find the solutions of each equation?
b. What mathematics concepts or principles did you apply in finding the solutions?
Explain how you applied these.
c. Compare your answers with those of your classmates. Did you arrive at the same
solutions? If NOT, explain.
Your goal in this section is to apply previously learned mathematics concepts
and principles in solving quadratic equations by extracting square roots. Use the
mathematical ideas and the examples presented in the preceding section to
Learn more about
Solving Quadratic
Equations by Extracting
Square Roots through
the WEB. You may
open the following links.
http://2012books.lardbucket.org/books/begi
nning-algebra/s12-01-extracting-square-
roots.html
http://www.purplemath.com/modules/solvqu
ad2.htm
http://www.personal.kent.edu/~bosikiew/Alg
ebra-handouts/solving-sqroot.pdf

DRAFT
March 24, 2014
26
Activity 7: What Does a Square Have?
Directions: Write a quadratic equation that represents the area of each square. Then
find the length of its side using the equation formulated. Answer the questions that
follow.
1. 2.
Questions:
a. How did you come up with the equation that represents the area of each shaded
region?
b. How did you find the length of side of each square?
c. Do all solutions to each equation represent the length of side of the square?
Explain your answer.
Was it easy for you to find the solutions of quadratic equations by extracting
square roots? Did you apply the different mathematics concepts and principles in
finding the solutions of each equation? I know you did!
In this section, the discussion was about solving quadratic equations by
extracting square roots.
Area = 169 cm2
s
s
Area = 256 cm2
s
s 5 cm
5 cm

DRAFT
March 24, 2014
27
Activity 8: Extract Then Describe Me!
Directions: Solve each of the following quadratic equations by extracting square
roots. Answer the questions that follow.
1. 123 2
t 4. 1502
x
2. 072
x 5.
16
92
x
3. 183 2
r 6.   0814 2
s
Questions:
a. How did you find the roots of each equation?
b. Which equation did you find difficult to solve by extracting square roots? Why?
c. Which roots are rational? Which are not? Explain your answer.
d. How will you approximate those roots that are irrational?
Activity 9: Intensify Your Understanding!
Directions: Answer the following.
1. Do you agree that a quadratic equation has at most two solutions? Justify your
answer and give examples.
2. Give examples of quadratic equations with (a) two real solutions, (b) one real
solution, and (c) no real solution.
3. Sheryl says that the solutions of the quadratic equations 492
w and
0492
w are the same. Do you agree with Sheryl? Justify your answer.
Your goal in this section is to take a closer look at some aspects of the topic.
You are going to think deeper and test further your understanding of solving
quadratic equations by extracting square roots. After doing the following
activities, you should be able to answer this important question: How does
finding solutions of quadratic equations facilitate in solving real-life problems and
in making decisions?
Were you able to find and describe the roots of each equation? Were you able to
approximate the roots that are irrational? I’m sure you did! Deepen further your
understanding of solving quadratic equations by extracting square roots by doing
the next activity.

DRAFT
March 24, 2014
28
4. Mr. Cruz asked Emilio to construct a square table such that its area is 3 m2. Is it
possible for Emilio to construct such table using an ordinary tape measure?
5. A 9 ft.2 square painting is mounted with border on a square frame. If the total area
of the border is 3.25 ft.2, what is the length of a side of the frame?
In this section, the discussion was about your understanding of solving quadratic
equations by extracting square roots.
What new realizations do you have about solving quadratic equations by
extracting square roots? How would you connect this to real life? How would you
use this in making decisions?

DRAFT
March 24, 2014
29
What to TRANSFER:
Activity 10: What More Can I Do?
1. Describe quadratic equations with 2 solutions, 1 solution, and no solution. Give at
least two examples for each.
2. Give at least five quadratic equations which can
be solved by extracting square roots, then solve.
3.
be given a practical task in which you will demonstrate your understanding of
solving quadratic equations by extracting square roots.
How did you find the performance task? How did the task help you see the real
world use of the topic?
Collect square tiles of different sizes. Using
these tiles, formulate quadratic equations that
can be solved by extracting square roots. Find
the solutions or roots of these equations.
This lesson was about solving quadratic equations by extracting square
roots. The lesson provided you with opportunities to describe quadratic equations
and solve these by extracting square roots. You were also able to find out how
such equations are illustrated in real life. Moreover, you were given the chance to
demonstrate your understanding of the lesson by doing a practical task. Your
understanding of this lesson and other previously learned mathematics concepts
and principles will enable you to learn about the wide applications of quadratic
equations in real life.

DRAFT
March 24, 2014
30
LEARNING MODULE
MATH 9
LESSON NO. 2B: SOLVING QUADRATIC EQUATIONS BY FACTORING
What to KNOW:
Activity 1: What Made Me?
Directions: Factor each of the following polynomials. Answer the questions that
follow.
1. xx 82 2
 6. 21102
 xx
2. ss 93 2
 7. 652
 xx
3. 2
204 xx  8. 25204 2
 rr
4. 2
105 tt  9. 49 2
t
5. 1282
 ss 10. 1432 2
 xx
Questions:
a. How did you factor each polynomial?
b. What factoring technique did you use to come up with the factors of each
polynomial? Explain how you used this technique.
c. How would you know if the factors you got are the correct ones?
d. Which of the polynomials did you find difficult to factor? Why?
Start Lesson 2B of this module by assessing your knowledge of the different
understanding solving quadratic equations by factoring. As you go through this
lesson, think of this important question: “How does finding solutions of quadratic
equations facilitate in solving real-life problems and in making decisions?” To
find the answer, perform each activity. If you find any difficulty in answering the
exercises, seek the assistance of your teacher or peers or refer to the modules
you have gone over earlier. You may check your answers with your teacher.

DRAFT
March 24, 2014
31
Activity 2: The Manhole
1. Draw a diagram to illustrate the given situation.
2. How are you going to represent the length and the
width of the pathway? How about its area?
3. What expression would represent the area of the cemented portion of the
pathway?
4. Suppose the area of the cemented portion of the pathway is 19.5 m2. What
equation would describe its area?
5. How will you find the length and the width of the pathway?
mathematics concepts or principles in factoring polynomials? I’m sure you were.
In the next activity, you will be representing a situation using a mathematical
sentence. This mathematical sentence will be used to satisfy the conditions of
the given situation.
A rectangular metal manhole with an area of
0.5 m2 is situated along a cemented pathway.
The length of the pathway is 8 m longer than
its width.
The activity you have just done shows how a real-life situation can be
represented by a mathematical sentence. Were you able to represent the given
situation by an equation? Do you now have an idea on how to use the equation in
finding the length and the width of the pathway? To further give you ideas in
solving the equation or other similar equations, perform the next activity.

DRAFT
March 24, 2014
32
Activity 3: Why is the Product Zero?
Directions: Use the equations below to answer the following questions.
1. How would you compare the three equations?
2. What value(s) of x would make each equation true?
3. How would you know if the value of x that you got satisfies each equation?
4. Compare the solutions of the given equations. What statement can you make?
5. Are the solutions of x + 7 = 0 and x – 4 = 0 the same as the solutions of
(x + 7)(x – 4) = 0? Why?
6. How would you interpret the meaning of the equation (x + 7)(x – 4) = 0?
Some quadratic equations can be solved easily by factoring. To solve such
quadratic equations, the following procedure can be followed.
1. Transform the quadratic equation into standard form if necessary.
2. Factor the quadratic expression.
3. Apply the zero product property by setting each factor of the quadratic expression
equal to 0.
quadratic equations by factoring? I’m sure you are!!! From the activities done,
you were able to find the factors of polynomials, represent a real-life situation by
a mathematical statement, and interpret zero product. But how does finding
solutions of quadratic equations facilitate in solving real-life problems and in
making decisions? You will find these out in the activities in the next section.
Before doing these activities, read and understand first some important notes on
solving quadratic equations by factoring and the examples presented.
x + 7 = 0 x – 4 = 0 (x + 7)(x – 4) = 0
Zero Product Property
If the product of two real numbers is zero, then either of the two is equal to
zero or both numbers are equal to zero.

DRAFT
March 24, 2014
33
4. Solve each resulting equation.
5. Check the values of the variable obtained by substituting each in the original
equation.
Example 1: Find the solutions of x2 + 9x = -8 by factoring.
a. Transform the equation into standard form ax2 + bx + c = 0.
x2 + 9x = -8  x2 + 9x + 8 = 0
b. Factor the quadratic expression x2 + 9x + 8.
x2 + 9x + 8 = 0  (x + 1)(x + 8) = 0
c. Apply the zero product property by setting each factor of the
quadratic expression equal to 0.
(x + 1)(x + 8) = 0  x + 1 = 0; x + 8 = 0
d. Solve each resulting equation.
x + 1 = 0  x + 1 – 1 = 0 – 1
x = -1
x + 8 = 0  x + 8 – 8 = 0 – 8
x = -8
e. Check the values of the variable obtained by substituting
each in the equation x2 + 9x = -8.
892
 xx is true when x = -1 or when x = -8.
 xx has two solutions: x = -1 or x = -8
For x = -1:
892
 xx
    8191
?
2

891
?

88 
For x = -8:
892
 xx
    8898
?
2

87264
?

88 

DRAFT
March 24, 2014
34
Example 2: Solve 9x2 – 4 = 0 by factoring.
To solve the equation, factor the quadratic expression 9x2 – 4.
9x2 – 4 = 0  (3x + 2)(3x – 2) = 0
Set each factor to 0.
3x + 2 = 0; 3x – 2 = 0
Solve each resulting equation.
Check the values of the variable obtained by substituting each in
the equation 9x2 – 4 = 0.
049 2
x is true when
3
2-
x or when
3
2
x .
Answer: The equation 049 2
x has two solutions:
3
2-
x or
3
2
x
For
3
2-
x :
049 2
x
?2
04
3
2
9 




 
?
04
9
4
9 





?
044 
00 
For
3
2
x :
049 2
x
?2
04
3
2
9 





?
04
9
4
9 





?
044 
00 
3x + 2 = 0  3x + 2 – 2 = 0 – 2
3x = -2
3
2
3
3 
=
x
3
2
x
3x – 2 = 0  3x – 2 + 2 = 0 + 2
3x = 2
3
2
=
3
3x
3
2
x

DRAFT
March 24, 2014
35
What to PROCESS:
Activity 4: Factor Then Solve!
Directions: Solve the following quadratic equations by factoring. Answer the
questions
that follow.
1. 072
 xx 6. xx 5142

2. 0186 2
 ss 7. 2
21511 rr 
3. 01682
 tt 8. 0252
x
4. 025102
 xx 9. 0481 2
 x
5. 1662
 hh 10. ss 1294 2

Questions:
d. How did you find the solutions of each equation?
e. What mathematics concepts or principles did you apply in finding the solutions?
f. Compare your answers with those of your classmates. Did you arrive at the same
and principles in solving quadratic equations by factoring. Use the mathematical
ideas and the examples presented in the preceding section to answer the
activities provided.
Learn more about
Solving Quadratic
Equations by Factoring
links.
nning-algebra/s09-06-solving-equations-
by-factoring.html
http://www.purplemath.com/modules/solvq
uad.htm
http://www.webmath.com/quadtri.html
http://www.mathwarehouse.com/quadratic/
solve-quadratic-equation-by-factoring.php

DRAFT
March 24, 2014
36
Activity 5: What Must Be My Length and Width?
Directions: The quadratic equation given describes the area of the shaded region of
each figure. Use the equation to find the length and width of the figure. Answer the
2. 2.
Questions:
a. How did you find the length and width of each figure?
b. Can all solutions to each equation be used to determine the length and width of
each figure? Explain your answer.
are going to think deeper and test further your understanding of solving quadratic
equations by factoring. After doing the following activities, you should be able to
answer this important question: How does finding solutions of quadratic
equations facilitate in solving real-life problems and in making decisions?
Was it easy for you to find the solutions of quadratic equations by factoring? Did
you apply the different mathematics concepts and principles in finding the
solutions of each equation? I know you did!
factoring.
(s + 6)(s – 2) = 153
(2s + 3)(s + 4) = 228s
s
2 4
s 6 2s 3

DRAFT
March 24, 2014
37
Activity 6: How Well Did I Understand?
Directions: Answer each of the following.
1. Which of the following quadratic equations may be solved more appropriately by
factoring? Explain your answer.
a. 722 2
x c. 0642
w
b. 036122
 tt d. 01082 2
 ss
2. Patricia says that it’s more appropriate to use the method of factoring than
extracting square roots in solving the quadratic equation 4x2 – 9 = 0. Do you
agree with Patricia? Explain your answer.
3. Do you agree that not all quadratic equations can be solved by factoring? Justify
your answer by giving examples.
4. Find the solutions of each of the following quadratic equations by factoring.
Explain how you arrived at you answer.
a. (x + 3)2 = 25 c. (2t – 3)2 = 2t2 + 5t – 26
b. (s + 4)2 = -2s d. 3(x + 2)2 = 2x2 + 3x – 8
5. Do you agree that x2 + 5x – 14 = 0 and 14 – 5x – x2 = 0 have the same solutions?
Justify your answer.
6. Show that the equation (x – 4)2 = 9 can be solved both by factoring and extracting
square roots.
7. A computer manufacturing company would like to come up with a new laptop
computer such that its monitor is 80 square inches smaller than the present ones.
Suppose the length of the monitor of the larger computer is 5 inches longer than
its width and the area of the smaller computer is 70 square inches. What are the
dimensions of the monitor of the larger computer?
equations by factoring.
What new insights do you have about solving quadratic equations by factoring?
How would you connect this to real life? How would you use this in making
decisions?

DRAFT
March 24, 2014
38
What to TRANSFER:
Activity 7: Meet My Demands!!!
Mr. Lakandula would like to increase his production of milkfish (bangus) due to its
high demand in the market. He is thinking of making a larger fishpond in his 8000 sq
m lot near a river. Help Mr. Lakandula by making a sketch plan of the fishpond to be
made. Out of the given situation and the sketch plan made, formulate as many
quadratic equations then solve by factoring. You may use the rubric in the next page
to rate your work.
Rubric for Sketch Plan and Equations Formulated and Solved
4 3 2 1
The sketch plan is
accurately made,
presentable, and
appropriate.
The sketch plan is
accurately made
and appropriate.
The sketch plan is
not accurately
made but
appropriate.
The sketch plan
is made but not
appropriate.
Quadratic
equations are
accurately
formulated and
solved correctly.
Quadratic
equations are
accurately
formulated but not
all are solved
correctly.
Quadratic
equations are
accurately
formulated but are
not solved
correctly.
Quadratic
equations are
accurately
formulated but
are not solved.
be given a practical task which will demonstrate your understanding of solving
quadratic equations by factoring.
This lesson was about solving quadratic equations by factoring. The
lesson provided you with opportunities to describe quadratic equations and solve
these by factoring. You were able to find out also how such equations are
illustrated in real life. Moreover, you were given the chance to demonstrate your
understanding of the lesson by doing a practical task. Your understanding of this
lesson and other previously learned mathematics concepts and principles will
facilitate your learning of the wide applications of quadratic equations in real life.

DRAFT
March 24, 2014
39
LEARNING MODULE
MATH 9
LESSON NO. 2C: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE
SQUARE
What to KNOW:
Activity 1: How Many Solutions Do I Have?
Directions: Find the solution/s of each of the following equations. Answer the
1. 1712 x 6.
2
1
4
3
x
2. 915 s 7.   3610 2
x
3. 1225 r 8.   129 2
w
4. 3
6
5
x 9.
16
9
2
1
2






k
5. 5
7
4
t 10.
2
1
5
3
2






h
Start Lesson 2C of this module by assessing your knowledge of the different
understanding Solving Quadratic Equations by Completing the Square. As you
go through this lesson, think of this important question: “How does finding
making decisions?” To find the answer, perform each activity. If you find any
difficulty in answering the exercises, seek the assistance of your teacher or
peers or refer to the modules you have gone over earlier. You may check your
answers with your teacher.

DRAFT
March 24, 2014
40
Questions:
a. How did you find the solution(s) of each equation?
b. Which of the equations has only one solution? Why?
c. Which of the equations has two solutions? Why?
d. Which of the equations has solutions that are irrational? Why?
e. Were you able to simplify those solutions that are irrational? Why?
f. How did you write those irrational solutions?
Activity 2: Perfect Square Trinomial to Square of a Binomial
Directions: Express each of the following perfect square trinomials as a square of a
binomial. Answer the questions that follow.
1. 442
 xx 6. 81182
 xx
2. 36122
 tt 7.
9
1
3
22
 tt
3. 25102
 ss 8.
4
49
72
 rr
4. 64162
 xx 9.
64
9
4
32
 ss
5. 49142
 hh 10.
4
25
52
 ww
mathematics concepts or principles in finding the solution/s of each equation?
I’m sure you did! In the next activity, you will be expressing a perfect square
trinomial as a square of a binomial. I know that you already have an idea on how
to do this. This activity will help you in solving quadratic equations by completing
the square.

DRAFT
March 24, 2014
41
Questions:
a. How do you describe a perfect square trinomial?
b. How did you express each perfect square trinomial as the square of a binomial?
c. What mathematics concepts or principles did you apply to come up with your
answer? Explain how you applied these.
d. Compare your answer with those of your classmates. Did you get the same
answer? If NOT, explain.
e. Observe the terms of each trinomial. How is the third term related to the
coefficient of the middle term?
f. Is there an easy way of expressing a perfect square trinomial as a square of a
binomial? If there is any, explain how.
Activity 3: Make It Perfect!!!
Directions: Determine a number that must be added to make each of the following a
perfect square trinomial. Explain how you arrived at your answer.
1. ____22
 xx 6. ____112
 xx
2. ____202
 tt 7. ____152
 xx
3. ____162
 rr 8. ____212
 ww
4. ____242
 rr 9. ____
3
22
 ss
5. ____302
 xx 10. ____
4
32
 hh
Were you able to express each perfect square trinomial as a square of a
binomial? I’m sure you did! Let us further strengthen your knowledge and skills in
mathematics particularly in writing perfect square trinomials by doing the next
activity.
Was it easy for you to determine the number that must be added to the terms of
a polynomial to make it a perfect square trinomial? Were you able to figure out
how it can be easily done? In the next activity, you will be representing a
situation using a mathematical sentence. Such a mathematical sentence will be
used to satisfy the conditions of the given situation.

DRAFT
March 24, 2014
42
Activity 4: Finish the Contract!
Extracting square roots and factoring are usually used to solve quadratic
equations of the form 02
 cax . If the factors of the quadratic expression of
02
 cbxax are determined, then it is more convenient to use factoring to solve
it.
Another method of solving quadratic equation is by completing the square.
This method involves transforming the quadratic equation ax2 + bx + c = 0 into the
form (x – h)2 = k, where k ≥ 0. Can you tell why the value of k should be positive?
To solve the quadratic equation ax2 + bx + c = 0 by completing the square,
the following steps can be followed:
1. Divide both sides of the equation by a then simplify.
2. Write the equation such that the terms with variables are on the left side of
the equation and the constant term is on the right side.
quadratic equations by completing the square? I’m sure you are!!! From the
activities done, you were able to solve equations, express a perfect square
trinomial as a square of a binomial, write perfect square trinomials, and represent
a real-life situation by a mathematical sentence. But how does finding solutions of
decisions? You will find these out in the activities in the next section. Before
doing these activities, read and understand first some important notes on Solving
Quadratic Equations by Completing the Square and the examples presented.
Directions: The shaded region of the
diagram at the right shows the portion of a
square-shaped car park that is already
cemented. The area of the cemented part is
600 m2. Use the diagram to answer the
following questions.
1. How would you represent the length of
the side of the car park? How about the
width of the cemented portion?
2. What equation would represent the area
of the cemented part of the car park?
3. Using the equation formulated, how are
you going to find the length of a side of
the car park?
10 m
10 m

DRAFT
March 24, 2014
43
3. Add the square of one-half of the coefficient of x on both sides of the
resulting equation. The left side of the equation becomes a perfect square
trinomial.
4. Express the perfect square trinomial on the left side of the equation as a
square of a binomial.
5. Solve the resulting quadratic equation by extracting the square root.
6. Solve the resulting linear equations.
7. Check the solutions obtained against the original equation.
Example 1: Solve the quadratic equation 2x2 + 8x – 10 = 0 by completing the
square.
Divide both sides of the equation by 2 then simplify.
2x2 + 8x – 10 = 0 
2
0
2
1082 2

 xx
x2 + 4x – 5= 0
Add 5 to both sides of the equation then simplify.
x2 + 4x – 5= 0  x2 + 4x – 5 + 5 = 0 + 5
x2 + 4x = 5
Add to both sides of the equation the square of one-half of 4.
x2 + 4x = 5  x2 + 4x + 4 = 5 + 4
x2 + 4x + 4 = 9
Express x2 + 4x + 4 as a square of a binomial.
x2 + 4x + 4 = 9  (x + 2)2 = 9
Solve (x + 2)2 = 9 by extracting the square root.
(x + 2)2 = 9  92 x
x + 2 = ±3
Solve the resulting linear equations.
x + 2 = 3
x + 2 – 2 = 3 – 2
x = 1
x + 2 = -3
x + 2 – 2 = -3 – 2
x = -5
  24
2
1
  22 = 4

DRAFT
March 24, 2014
44
Check the solutions obtained against the original equation
2x2 + 8x – 10 = 0.
01082 2
 xx is true when x = 1 or when x = -5.
 xx has two solutions:
x = 1 or x = -5
Example 2: Find the solutions of the equation 01832
 xx by completing
the square.
01832
 xx  180181832
 xx
1832
 xx
Add to both sides of the equation the square of one-half of 3.
1832
 xx 
4
9
18
4
9
32
 xx
4
9
4
72
4
9
32
 xx 
4
81
4
9
32
 xx
Express
4
9
32
 xx as a square of a binomial.
4
81
4
9
32
 xx 
4
81
2
3
2






x
Solve
4
81
2
3
2






x by extracting the square root.
For x = 1:
01082 2
 xx
   
?
2
0101812 
 
?
010812 
?
01082 
00 
For x = -5:
01082 2
 xx
   
?
2
0105852 
 
?
01040252 
?
0104050 
00 
 
2
3
3
2
1
 
4
9
2
3
2







DRAFT
March 24, 2014
45
4
81
2
3
2






x 
4
81
2
3
x
2
9
2
3
x
Check the solutions obtained against the equation 01832
 xx .
01832
 xx is true when x = 3 or when x = -6.
 xx has two solutions: x = 3 or x = -6
Example 3: Find the solutions of x2 – 6x – 41 = 0 by completing the square.
x2 – 6x – 41 = 0  x2 – 6x – 41 + 41 = 0 + 41
x2 – 6x = 41
2
6
2
3
2
9
2
3
2
3
2
9
2
3



x
x
x
x = 3
2
12
2
3
2
9
2
3
2
3
2
9
2
3



x
x
x
x = -6
For x = 3:
01832
 xx
   
?
2
018333 
?
01899 
00 
For x = -6:
01832
 xx
   
?
2
018636 
?
0181836 
00 

DRAFT
March 24, 2014
46
Add to both sides of the equation the square of one-half of -6.
x2 – 6x = 41  x2 – 6x + 9 = 41 + 9
x2 – 6x + 9 = 50
Express x2 – 6x + 9 as a square of a binomial.
x2 – 6x + 9 = 50  (x – 3)2 = 50
Solve (x – 3)2 = 50 by extracting the square root.
(x – 3)2 = 50  503 x
Hence, 503 x is the same as 253 x .
Check the solutions obtained against the equation
x2 – 6x – 41 = 0.
253 x
32533 x
253 x
For 253 x :
04162
 xx
    0412536253
?2

04123018502309
?

00 
  36
2
1
  (-3)2 = 9
253 x
32533 x
253 x
50 can be expressed as 225  or 225  .
Notice that 25 is a perfect square. So, 225  can
be simplified further to 25 .

DRAFT
March 24, 2014
47
04062
 xx is true when 253 x or when
253 x .
 xx has two solutions:
253 x or 253 x
What to PROCESS:
Your goal in this section is to apply the key concepts of solving quadratic
equations by completing the square. Use the mathematical ideas and the
examples presented in the preceding section to answer the activities provided.
Learn more about
Solving Quadratic
Equations by
links.
http://www.purplemath.com/modules/sqrqu
ad.htm
nning-algebra/s12-02-completing-the-
square.html
http://www.regentsprep.org/Regents/math/a
lgtrig/ATE12/indexATE12.htm
http://www.mathsisfun.com/algebra/complet
ing-square.html
For 253 x :
04162
 xx
    0412536253
?2

04123018502309
?

00 

DRAFT
March 24, 2014
48
Activity 5: Complete Me!
Directions: Find the solutions of each of the following quadratic equations by
completing the square. Answer the questions that follow.
1. x2 – 2x = 3 6. 4x2 – 32x = -28
2. s2 + 4s – 21 = 0 7. 0652
 xx
3. t2 + 10t + 9 = 0 8. 0
4
51
72
 mm
4. x2 + 14x = 32 9. r2 + 4r = -1
5. r2 – 10r = -17 10. w2 + 6w – 11 = 0
Questions:
b. What mathematics concepts or principles did you apply in finding the solutions?
Was it easy for you to find the solutions of quadratic equations by completing
the square? Did you apply the different mathematics concepts and principles in
finding the solutions of each equation? I know you did!

DRAFT
March 24, 2014
49
Activity 6: Represent then Solve!
Directions: Using each figure, write a quadratic equation that represents the area of
the shaded region. Then find the solutions to the equation by completing the square.
Answer the questions that follow.
1. 2.
Questions:
a. How did you come up with the equation that represents the area of each shaded
region?
b. How did you find the solution/s of each equation?
c. Do all solutions to each equation represent a particular measure of each figure?
Area = 88
cm2
3 cm
s
s
5 cm
Area = 176
cm2
t
t
Were you able to come up with the right representations of the area of the
shaded region of each figure? Were you able to solve the equations formulated
and obtain the appropriate measure that would describe each figure?
completing the square.

DRAFT
March 24, 2014
50
Activity 7: What Solving Quadratic Equations by Completing the Square Means
to Me…
1. Karen wants to use completing the square in solving the quadratic equation
4x2 – 25 = 0. Can she use it in finding the solutions of the equation? Explain why
or why not.
2. Do you agree that any quadratic equation can be solved by completing the
square? Justify your answer.
3. If you are to choose between completing the square and factoring in finding the
solutions of each of the following equations, which would you choose? Explain
and solve the equation using your preferred method.
a. 4x2 – 20x = 11 b. x2 + 7x + 12 = 0
4. Gregorio solved the equation 2x2 – 8x + 15 = 0. The first few parts of his solution
are shown below.
2x2 – 8x + 15 = 0  2x2 – 8x + 15 – 15 = 0 – 15
2x2 – 8x = –15
2x2 – 8x + 16 = –15 + 16
Do you think Gregorio arrived at the correct solutions of the equation? Justify
your answer.
5. An open box is to be formed out of a rectangular piece of cardboard whose
length is 8 cm longer than its width. To form the box, a square of side 4 cm will be
removed from each corner of the cardboard. Then the edges of the remaining
cardboard will be turned up.
a. Draw a diagram to illustrate the given situation.
b. How would you represent the dimensions of the cardboard?
c. What expressions represent the length, width, and height of the box?
  48
2
1
 ; (-4)2 = 16
equations by completing the square. After doing the following activities, you
should be able to answer this important question: How does finding solutions of
decisions?

DRAFT
March 24, 2014
51
d. If the box is to hold 448 cm3, what mathematical sentence would represent the
given situation?
e. Using the mathematical sentence formulated, how are you going to find the
dimensions of the rectangular piece of cardboard?
f. What are the dimensions of the rectangular piece of cardboard?
g. What is the length of the box? How about its width and height?
6. From 2006 through 2012, the average weekly income of an employee in a certain
company is estimated by the quadratic expression 324048.618.0 2
 nn , where
n is the number of years after 2006. In what year did the average weekly income
of an employee become Php3268.80?
What to TRANSFER:
Activity 8: Design Packaging Boxes!!!
Directions: Perform the following.
A. Designing Open Boxes
1. Make sketch plans of 5 rectangular open boxes such that:
a. the heights of the boxes are the same; and
b. the boxes can hold 240 cm3, 270 cm3, 504 cm3, 810 cm3, and 468 cm3,
respectively.
2. Write a quadratic equation that would represent the volume of each box.
be given a practical task which will demonstrate your understanding of solving
quadratic equations by completing the square.
equations by completing the square.
What new insights do you have about solving quadratic equations by completing
the square? How would you connect this to real life? How would you use this in
making decisions?

DRAFT
March 24, 2014
52
3. Solve each quadratic equation by completing the square to determine the
dimensions of the materials to be used in constructing each box.
B. Designing Covers of the Open Boxes
1. Make sketch plans of covers of the open boxes in Part A such that:
a. the heights of the covers are the same; and
b. the base of each cover is rectangular.
2. Write a quadratic equation that would represent the volume of the box’s cover.
3. Solve each quadratic equation by completing the square to determine the
dimensions of the materials to be used in constructing the box’s cover.
4 3 2 1
The sketch plan is
accurately made,
presentable, and
appropriate.
The sketch plan is
accurately made
and appropriate.
The sketch plan is
not accurately
made but
appropriate.
The sketch plan
is made but not
appropriate.
Quadratic
equations are
accurately
formulated and
solved correctly.
Quadratic
equations are
accurately
formulated but not
all are solved
correctly.
Quadratic
equations are
accurately
formulated but are
not solved
correctly.
Quadratic
equations are
accurately
formulated but
are not solved.
This lesson was about solving quadratic equations by completing the
square. The lesson provided you with opportunities to describe quadratic
equations and solve these by completing the square. You were able to find out also
how such equations are illustrated in real life. Moreover, you were given the
chance to demonstrate your understanding of the lesson by doing a practical task.
Your understanding of this lesson and other previously learned mathematics
concepts and principles will facilitate your learning of the wide applications of
quadratic equations in real life.

DRAFT
March 24, 2014
53
LEARNING MODULE
MATH 9
LESSON NO. 2D: SOLVING QUADRATIC EQUATIONS BY USING THE
QUADRATIC FORMULA
What to KNOW:
Activity 1: It’s Good to be Simple!
Directions: Work with a partner in simplifying each of the following expressions.
1.
)3(2
96 
6.
)1(2
20366 
2.
)3(2
96 
7.
)4(2
100255 
3.
)2(2
186 
8.
)3(2
521010 2

4.
)2(2
249 
9.
)4(2
1644 2

5.
)3(2
28648


10.
)3(2
)2)(3(455 2

Start Lesson 2D of this module by assessing your knowledge of the different
understanding solving quadratic equations by using the quadratic formula. As
you go through this lesson, think of this important question: “How does finding
making decisions?” To find the answer, perform each activity. If you find any
difficulty in answering the exercises, seek the assistance of your teacher or
peers or refer to the modules you have gone over earlier. You may check your
answers with your teacher.

DRAFT
March 24, 2014
54
Questions:
a. How would you describe the expressions given?
b. How did you simplify each expression?
c. Which expression did you find difficult to simplify? Why?
d. Compare your work with those of your classmates. Did you arrive at the same
answer?
Activity 2: Follow the Standards!
Directions: Write the following quadratic equations in standard form, ax2 + bx + c = 0.
Then identify the values of a, b, and c. Answer the questions that follow.
1. 2x2 + 9x =10 6. x(5 – 2x) + 15 = 0
2. -2x2 = 2 – 7x 7. (x + 4)(x + 12) = 0
3. 6x – 1 = 2x2 8. (x – 6)(x – 9) = 0
4. 10 + 7x – 3x2 = 0 9. (3x + 7)(x – 1) = 0
5. 2x(x – 6) = 5 10. 3(x – 5)2 + 10 = 0
Questions:
b. How do you describe a quadratic equation that is written in standard form?
c. Are there different ways of writing a quadratic equation in standard form? Justify
your answer.
How did you find the activity? Were you able to simplify the expressions? I’m
sure you did! In the next activity, you will be writing quadratic equations in
standard form. You need this skill for you to solve quadratic equations by using
the quadratic formula.
Were you able to write each quadratic equation in standard form? Were you able
to determine the values of a, b, and c? If you did, then extend further your
understanding of the real-life applications of quadratic equations by doing the
next activity.

DRAFT
March 24, 2014
55
Activity 3: Why do the Gardens Have to be Adjacent?
1. How would you represent the dimensions of each garden?
2. What mathematical sentence would represent the length of fencing material to be
used in enclosing the two gardens?
How about the mathematical sentence that would represent the total area of the
two gardens?
3. How will you find the dimensions of each garden?
4. What equation will you use in finding the dimensions of each garden?
5. How would you describe the equation formulated in item 4?
How are you going to find the solutions of this equation?
6. Do you think the methods of solving quadratic equations that you already learned
can be used to solve the equation formulated in item 4? Why?
Mr. Bonifacio would like to enclose
his two adjacent rectangular gardens
with 70.5 m of fencing materials. The
gardens are of the same size and
their total area is 180 m2.
Did the activity you just performed capture your interest? Were you able to
formulate mathematical sentence that will lead you in finding the measures of the
unknown quantities? In the next activity, you will be given the opportunity to
derive a general mathematical sentence which you can use in solving quadratic
equations.

DRAFT
March 24, 2014
56
Activity 4: Lead Me to the Formula!
Directions: Work in groups of 4 in finding the solutions of the following quadratic
equation by completing the square. Answer the questions that follow.
1. How did you use completing the square in solving the given equation? Show the
complete details of your work.
2. What are the solutions of the given equation?
3. How would you describe the solutions obtained?
4. Compare your work with those of other groups. Did you obtain the same
5. In the equation 2x2 + 9x + 10 = 0, what would be the resulting equation if 2, 9,
and 10 were replaced by a, b, and c, respectively?
6. Using the resulting equation in item 5, how are you going to find the value of x if
you follow the same procedure in finding the solutions of 2x2 + 9x + 10 = 0?
What equation or formula would give the value of x?
7. Do you think the equation or formula that would give the value of x can be used in
solving other quadratic equations? Justify your answer by giving examples.
2x2
+ 9x + 10 = 0
quadratic equations by using the quadratic formula? I’m sure you are!!! From the
activities done, you were able to solve equations, express a perfect square
trinomial as a square of a binomial, write perfect square trinomials, and
represent a real-life situation by a mathematical sentence. But how does finding
making decisions? You will find these out in the activities in the next section.
Before doing these activities, read and understand first some important notes on
Solving Quadratic Equations by using the Quadratic Formula and the examples
presented.

DRAFT
March 24, 2014
57
The solutions of any quadratic equation ax2 + bx + c = 0 can be determined
using the quadratic formula
2a
4acbb 2

x , a ≠ 0. This formula can be derived
by applying the method of completing the square as shown on the next page.
ax2 + bx + c = 0  ax2 + bx = -c Why?
a
c
a
ba 2


 xx

a
c
a
b2 

x
x
Why?
2a
b
a
b
2
1





 ;
2
22
4a
b
2a
b





 Why?
2
2
2
2
2
4a
b
a
c
4a
b
a
b



x
x
Why?
2
2
2
22
4a
4acb
4a
b4ac
2a
b 








x
Why?
2
2
4a
4acb
2a
b 
x 
2a
4acb
2a
b 2

x
Why?
2a
b
2a
4acb2


x
Why?
2a
4acbb 2

x
Why?
To solve any quadratic equation ax2 + bx + c = 0 using the quadratic formula,
determine the values of a, b, and c, then substitute these in the equation
2a
4acbb 2

x . Simplify the result if possible then check the solutions obtained
against the original equation.
Example 1: Find the solutions of the equation 2x2 + 3x = 27 using the
quadratic formula.
Write the equation in standard form.
2x2 + 3x = 27 2x2 + 3x – 27 = 0
Determine the values of a, b, and c.
2x2 + 3x – 27 = 0  a = 2; b = 3; c = -27
Substitute the values of a, b, and c in the quadratic formula.

DRAFT
March 24, 2014
58
2a
4acbb 2

x 
      
 22
27-2433 2

x
Simplify the result.
      
 22
27-2433 2

x 
4
21693 
x
4
2253 
x
4
153 
x
Check the solutions obtained against the equation 2x2 + 3x = 27.
When x = 3:
    273332
?
2
      273392
?

27918
?

2727 
When
2
9
x :
27
2
9
3
2
9
2
?2












  27
2
9
3
4
81
2
?












27
2
27
2
81 ?

27
2
54 ?

2727 
2732 2
 xx is true when x = 3 or when
2
9
x .
4
12
4
153


x
x = 3
4
18
4
153 


x
2
9
x

DRAFT
March 24, 2014
59
 xx has two solutions: x = 3 or
2
9
x .
What to PROCESS:
Activity 5: Is the Formula Effective?
Directions: Find the solutions of each of the following quadratic equations using the
quadratic formula. Answer the questions that follow.
1. 09102
 xx 6. 0972 2
 xx
2. 035122
 xx 7. 0144 2
 xx
3. 01452
 xx 8. 043 2
 xx
4. 01242
 xx 9. 0729 2
x
5. 472
 xx 10. 342 2
 xx
Questions:
a. How did you use the quadratic formula in finding the solution/s of each equation?
b. How many solutions does each equation have?
Learn more about
Solving Quadratic
Equations by Using the
Quadratic Formula
links.
nning-algebra/s12-03-quadratic-
formula.html
http://www.regentsprep.org/Regents/math/a
lgtrig/ATE3/quadformula.htm
http://www.purplemath.com/modules/quadf
orm.htm
http://www.algebrahelp.com/lessons/equati
ons/quadratic/
Your goal in this section is to apply the key concepts of solving quadratic
equations by using the quadratic formula. Use the mathematical ideas and the
examples presented in the preceding section to answer the activities provided.

DRAFT
March 24, 2014
60
c. Is there any equation whose solutions are equal? If there is any, describe the
equation.
d. Is there any equation with zero as one of the solutions? Describe the equation if
there is any.
e. Compare your answers with those of your classmates. Did you arrive at the same
Activity 6: Cut to Fit!
Directions: Read and understand the situation below then answer the questions that
follow.
Mr. Bonifacio cuts different sizes of rectangular plywood to be used in the
furniture that he makes. Some of these rectangular plywood are described below.
Plywood 1: The length of the plywood is twice its width and the area is 4.5
sq ft.
Plywood 2: The length of the plywood is 1.4 ft. less than twice its width and
the
area is 16 sq ft.
Plywood 3: The perimeter of the plywood is 10 ft. and its area is 6 sq ft.
Questions:
1. What quadratic equation represents the area of each piece of plywood? Write the
equation in terms of the width of the plywood.
2. Write each quadratic equation formulated in item 1 in standard form. Then
determine the values of a, b, and c.
3. Solve each quadratic equation using the quadratic formula.
4. Which of the solutions or roots obtained represents the width of each plywood?
5. What is the length of each piece of plywood? Explain how you arrived at your
answer.
Was it easy for you to find the solutions of quadratic equations by using the
quadratic formula? Were you able to simplify the solutions obtained? I know you
did!

DRAFT
March 24, 2014
61
Activity 7: Make the Most Out of It!
1. The values of a, b, and c of a quadratic equation written in standard form are -2,
8, and 3, respectively. Another quadratic equation has 2, -8, and -3 as the values
of a, b, and c, respectively. Do you agree that the two equations have the same
solutions? Justify your answer.
2. How are you going to use the quadratic formula in determining whether a
quadratic equation has no real solutions? Give at least two examples of quadratic
equations with no real solutions.
3. Find the solutions of the following quadratic equations using the quadratic
formula. Tell whether the solutions are real numbers or not real numbers. Explain
your answer.
a. x2 + 2x + 9 = 0 c. (2x – 5)2 – 4 = 0
b. 2x2 + 4x + 7 = 0 d. (3x + 2)2 = 3x + 10
4. Do you think the quadratic formula is more appropriate to use in solving quadratic
equations? Explain then give examples to support your answer.
equations by using the quadratic formula. After doing the following activities, you
should be able to answer this important question: How does finding solutions of
decisions?
Were you able to come up with the quadratic equation that represents the area
of each piece of plywood? Were you able to determine the length and width of
each piece of plywood?
In this section, the discussion was about solving quadratic equations by using
the quadratic formula.

DRAFT
March 24, 2014
62
5. If you are to solve each of the following quadratic equations, which method would
you use and why? Explain your answer.
a. 9x2 = 225 d. 2x2 + x – 28 = 0
b. 4x2 – 121 = 0 e. 4x2 + 16x + 15 = 0
c. x2 + 11x + 30 = 0 f. 4x2 + 4x – 15 = 0
6. The length of a car park is 120 m longer than its width. The area of the car park is
6400 m2.
a. How would you represent the width of the car park?
How about its length?
b. What equation represents the area of the car park?
c. How would you use the equation representing the area of the car park in
finding its length and width?
d. What is the length of the car park? How about its width? Explain how you
arrived at your answer.
e. Suppose the area of the car park is doubled, would its length and width also
double? Justify your answer.
7. The length of a rectangular table is 0.6 m more than twice its width and its area is
4.6 m2. What are the dimensions of the table?
8. Grace constructed an open box with a square base out of 192 cm2 material. The
height of the box is 4 cm. What is the length of the side of the base of the box?
9. A car travels 30 kph faster than a truck. The car covers 540 km in three hours
less than the time it takes the truck to travel the same distance. What is the
speed of the car? What about the truck?
equations by using the quadratic formula.
What new insights do you have about solving quadratic equations by using the
quadratic formula? How would you connect this to real life? How would you use
this in making decisions?

DRAFT
March 24, 2014
63
What to TRANSFER:
Activity 8: Show Me The Best Floor Plan!!!
Mr. Luna would like to construct a new house with a floor area of 72 m2. He
asked an architect to prepare a floor plan that shows the following:
c. 2 bedrooms d. Comfort room
d. Living room e. Kitchen
e. Dining room f. Laundry Area
1. Suppose you were the architect asked by Mr. Luna to prepare a floor plan. How
will you do it? Draw the floor plan.
2. Formulate as many quadratic equations using the floor plan that you prepared.
Solve the equations using the quadratic formula.
4 3 2 1
The sketch plan is
accurately made,
presentable, and
appropriate.
The sketch plan is
accurately made
and appropriate.
The sketch plan is
not accurately
made but
appropriate.
The sketch plan
is made but not
appropriate.
Quadratic
equations are
accurately
formulated and
solved correctly.
Quadratic
equations are
accurately
formulated but not
all are solved
correctly.
Quadratic
equations are
accurately
formulated but are
not solved
correctly.
Quadratic
equations are
accurately
formulated but
are not solved.
be given a practical task which will demonstrate your understanding.

DRAFT
March 24, 2014
64
This lesson was about solving quadratic equations by using the quadratic
formula. The lesson provided you opportunities to describe quadratic equations
and solve these by using the quadratic formula. You were able to find out also how
such equations are illustrated in real life. Moreover, you were given the chance to
demonstrate your understanding of the lesson by doing a practical task. Your
understanding of this lesson and other previously learned mathematics concepts
and principles will facilitate your learning of the wide applications of quadratic
equations in real life.

DRAFT
March 24, 2014
65
LEARNING MODULE
MATH 9
LESSON NO. 3: THE NATURE OF THE ROOTS OF A QUADRATIC EQUATION
What to KNOW:
Activity 1: Which are Real? Which are Not?
Directions: Refer to the numbers below to answer the questions that follow.
I L O V E
M A T H
Start lesson 3 of this module by assessing your knowledge of the different
understanding the nature of roots of quadratic equations. As you go through this
lesson, think of this important question: “How does the nature of roots of
quadratic equation facilitate in understanding the conditions of real life
situations?” To find the answer, perform each activity. If you find any difficulty in
answering the exercises, seek the assistance of your teacher or peers or refer to
the modules you have gone over earlier. You may check your answers with your
teacher.
24. 5 289

DRAFT
March 24, 2014
66
Questions:
1. Which of the numbers above are familiar to you? Why? Describe these numbers.
2. Which of the numbers are real? Which are not real?
3. Which of the numbers are rational? irrational? Explain your answer.
4. Which of the numbers are perfect squares? not perfect squares?
5. How do you describe numbers that are perfect squares?
Activity 2: Math in A, B, C?
Directions: Write the following quadratic equations in standard form, ax² + bx + c = 0,
then identify the values of a, b, and c. Answer the questions that follow.
ax² + bx + c = 0
1. x² + 5x = 4 _______________ a = ____ b = ____ c = _____
2. -4x2 = 8x – 3 _______________ a = ____ b = ____ c = _____
3. 10x – 1 = 4x² _______________ a = ____ b = ____ c = _____
4. 15 + 8x – 3x2 = 0 _______________ a = ____ b = ____ c = _____
5. 3x(x – 14) = 12 _______________ a = ____ b = ____ c = _____
Questions:
b. Aside from your answer, do you think there is another way of writing each
quadratic equation in standard form? If YES, show then determine the values of
a, b, and c.
Were you able to classify the given numbers as real or not real, and as
rational or irrational? In the next activity, you will determine the values of a, b,
and c when the quadratic equation is written in the form ax² + bx + c = 0. You
have done this activity in the previous lessons so I am sure you are already
familiar with this.
Were you able to rewrite the given quadratic equations in the form
ax²+bx+c = 0? Were you able to find the value of a, b and c? In the next activity,
you will find the value of b2 – 4ac. This value will be your basis in describing the
roots of a quadratic equation.

DRAFT
March 24, 2014
67
Activity 3: What’s My Value?
Directions: Evaluate the expression b² – 4ac given the following values of a, b, and c.
1. a = 1, b = 5, c = 4
2. a = 2, b = 1, c = -21
3. a = 4, b = 4, c = 1
4. a = 1, b = - 2, c = -2
5. a = 9, b = 0, c = 16
Activity 4: Find my Equation and Roots
Directions: Using the values of a, b and c in Activity 3, write the quadratic equation
ax² + bx + c = 0.Then find the roots of each resulting equation.
ax² + bx +c = 0 Roots
1. _________________________ _________________
2. _________________________ _________________
3. _________________________ _________________
4. _________________________ _________________
5. _________________________ _________________
Were you able to evaluate the expression b2 – 4ac given the values of a,
b and c? What do you think is the importance of the expression b2 – 4ac in
determining the nature of the roots of quadratic equation? You will find this out
as you perform the succeeding activities.
Were you able to write the quadratic equation given the values of a, b and
c? Were you able to find the roots of the resulting quadratic equation? In the
next activity, you will describe the nature of the roots of quadratic equation using
the value of b2 – 4ac.

DRAFT
March 24, 2014
68
Activity 5: Place Me on the Table!
1. Complete the table below using your answers in activities 3 and 4.
Equation b² - 4ac Roots
1.
2.
3.
4.
5.
2. How would you describe the roots of quadratic equation when the value of b2 –
4ac is 0? positive and perfect square? positive but not perfect square? negative?
3. Which quadratic equation has roots that are real numbers and equal? rational
numbers? Irrational numbers? not real numbers?
4. How do you determine the quadratic equation having roots that are real numbers
and equal? rational numbers? irrational numbers? not real numbers?
Activity 6: Let’s Shoot that Ball!
Directions: Study the situation below and answer the questions that follow.
A basketball player throws a ball vertically with an initial velocity of 100ft/sec.
The distance of the ball from the ground after t seconds is given by the expression
100t – 16t².
1. What is the distance of the ball from the ground after 6 seconds?
2. After how many seconds does the ball reach a distance of 50 ft. from the ground?
3. How many seconds will it take for the ball to fall to the ground?
4. Do you think the ball can reach the height of 160 ft.? Why? Why not?
Were you able to relate the value of b² - 4ac to the nature of the roots of
the quadratic equation? In the next activity, you will find out how the discriminant
of the quadratic equation is illustrated in real-life situations.

DRAFT
March 24, 2014
69
The value of the expression b2 – 4ac is called the discriminant of the
quadratic equation ax² + bx + c = 0. This value can be used to describe the nature of
the roots of a quadratic equation. It can be zero, positive and perfect square, positive
but not perfect square or negative.
1. When b2 – 4ac is equal to zero, then the roots are real numbers and are equal.
Example: Describe the roots of 0442
 xx .
The values of a, b, and c in the equation are the following.
a = 1 b = -4 c = 4
Substitute these values of a, b and c in the expression b² – 4ac.
    4144ac4b 22

1616 
0
Since the value of b2 – 4ac is zero, we can say that the roots of
the quadratic equation 0442
 xx are real numbers and are
equal.
This can be checked by determining the roots of
0442
 xx using any of the methods of solving quadratic
equation.
How did you find the preceding activities? Are you ready to learn more about
the nature of the roots of quadratic equations? From the activities you have
done, you were able to determine the nature of roots of quadratic equations,
whether they are real numbers, not real numbers, rational or irrational numbers.
Moreover, you were able to find out how quadratic equations are illustrated in
real life. Find out more about the applications of quadratic equations by
performing the activities in the next section. Before doing these activities, read
and understand first some important notes on Quadratic Equations and the
examples presented.

DRAFT
March 24, 2014
70
If the quadratic formula is used, the roots that can be obtained
are the following.
The roots of the quadratic equation 0442
 xx are real
numbers and are equal.
2. When b2 – 4ac is greater than zero and a perfect square, then the roots are
rational numbers but are not equal.
Example: Determine the nature of the roots of 01072
 xx .
In the equation, the values of a, b, and c are 1, 7, and 10,
respectively. Use these values to evaluate b² – 4ac.
    10147ac4b 22

4049 
9
Since the value of b2 – 4ac is greater than zero and a perfect
square, then the roots of the quadratic equation
01072
 xx are rational numbers but are not equal.
To check, solve for the roots of 01072
 xx .
 xx are rational
numbers but are not equal.
      
 12
41444 2

x
2
04
2
16164 


x
2
4
2
04


x
2x
      
 12
41444 2

x
2
04
2
16164 


x
2
4
2
04


x
2x
2
2
4
2
37
2
97







x
x
x
5
2
10
2
37
2
97







x
x
x

DRAFT
March 24, 2014
71
3. When b2 – 4ac is greater than zero but not a perfect square, then the roots are
irrational numbers and are not equal.
Example: Describe the roots of 0362
 xx .
Evaluate the expression b² – 4ac using the values a, b, and c.
respectively.
    3146ac4b 22

1236 
24
Since the value of b2 – 4ac is greater than zero but not a perfect
square, then the roots of the quadratic equation
0362
 xx are irrational numbers and are not equal.
 xx .
 xx are irrational
numbers and are not equal.
4. When b2 – 4ac is less than zero, then the equation has no real roots.
Example: Determine the nature of the roots of 0522
 xx .
respectively. Use these values to evaluate b2 – 4ac.
    5142ac4b 22

204 
16
Since the value of b2 – 4ac is less than zero, then the quadratic
equation 0522
 xx has no real roots.
63
2
626
2
246





x
x
63
2
626
2
246





x
x

DRAFT
March 24, 2014
72
 xx .
 xx are not real
numbers.
What to PROCESS:
  
 12
51422 2

x
2
2042 
x
2
162 
x
  
 12
51422 2

x
2
2042 
x
2
162 
x
Now that you have learned about the discriminant and how it determines the
nature of the roots of a quadratic equation, you are ready to perform the
succeeding activities.
Your goal in this section is to apply the key concepts of the discriminant of the
quadratic equation. Use the mathematical ideas and examples presented in the
preceding section to answer the activities provided.
Learn more about
the Nature of the Roots
of Quadratic Equations
links.
http://www.analyzemath.com/Equations/Qu
adratic-1.html
http://www.regentsprep.org/Regents/math/
algtrig/ATE3/discriminant.htm
http://www.icoachmath.com/math_dictionar
y/discriminant.html

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March 24, 2014
73
Activity 7: What is My Nature?
Directions: Determine the nature of the roots of the following quadratic equations
using the discriminant. Answer the questions that follow.
1. x2 + 6x + 9 = 0 discriminant : ______ nature of the roots: _________
2. x2 + 9x + 20 = 0 discriminant: ______ nature of the roots: _________
3. 2x2 – 10x + 8 = 0 discriminant: ______ nature of the roots: _________
6. 2x2 + 6x + 4 = 0 discriminant: ______ nature of the roots: _________
7. 3x2 – 5x = -4 discriminant: ______ nature of the roots: _________
8. 9x2 – 6x = -9 discriminant: ______ nature of the roots: _________
9. 10x2 – 4x = 8 discriminant: ______ nature of the roots: _________
10.3x² – 2x – 5 = 0 discriminant: ______ nature of the roots: _________
Questions:
a. How did you determine the nature of the roots of each quadratic equation?
b. When do you say that the roots of a quadratic equation are real or not real
numbers? rational or irrational numbers? equal or not equal?
c. How does the knowledge of the discriminant help you in determining the nature of
the roots of any quadratic equation?
Activity 8: Let’s Make a Table!
Mang Jose wants to make a table which has an area of 6 m2. The length of
the table has to be 1 m longer than the width.
a. If the width of the table is p meters, what will be its length?
b. Form a quadratic equation that represents the situation.
c. Without actually computing for the roots, determine whether the dimensions of
the table are rational numbers. Explain.
d. Give the dimensions of the table.
Were you able to determine the nature of the roots of any quadratic equation? I
know you did!

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March 24, 2014
74
What to REFLECT ON and FURTHER UNDERSTAND
Activity 9: How Well Did I Understand the Lesson?
Directions: Answer the following questions
1. Describe the roots of a quadratic equation when the discriminant is
a. zero. c. positive but not perfect square.
b. positive perfect square. d. negative.
Give examples for each.
2. How do you determine the nature of the roots of quadratic equation?
3. Danica says that the quadratic equation 2x2 + 5x – 4 = 0 has two possible
solutions because the value of its discriminant is positive. Do you agree with
Danica? Justify your answer.
4. When the quadratic expression ax² + bx + c is a perfect square trinomial, do you
agree that the value of its discriminant is zero? Justify your answer by giving at
least two examples.
Was it easy for you to determine the nature of the roots of the quadratic
equation? Try to compare your initial ideas with the discussion in this section.
How much of your initial ideas were found in this section? Which ideas are
different and need revision?
Now that you know the important ideas about the topic, let’s go deeper by
moving on to the next section. In the next section, you will develop further your
understanding of the nature of the roots of quadratic equations.
are going to think deeper and test further your understanding of the nature of the
roots of quadratic equations. After doing the following activities, you should be
able to answer this important question: How is the concept of the discriminant of
a quadratic equation used in solving real-life problems?

DRAFT
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75
5. You and a friend are camping. You want to hang your food pack from a branch
20 ft. from the ground. You will attach a rope to a stick and throw it over the
branch. Your friend can throw the stick upward with an initial velocity of 29 feet
per second. The distance of the stick after t seconds from an initial height of 6
feet is given by the expression 62916 2
 tt .
a. Form and describe the equation representing the situation. How did you come
up with the equation?
b. With the given conditions, will the stick reach the branch when thrown? Justify
your answer.
What to TRANSFER:
Activity 10: Will It or Will It Not?
Direction: Answer the following.
a. What will be the height of the ball after 2 seconds?
In this section, the discussion was about your understanding of the nature of the
roots of quadratic equations.
What new insights do you have about the nature of the roots of quadratic
equations? How would you connect this to real life? How would you use this in
making decisions?
1. When a basketball player shoots a ball from his
hand at an initial height of 2 m with an initial
upward velocity of 10 meters per second, the
height of the ball can be modeled by the quadratic
expression -4.9t² + 10t + 2 after t seconds.
Your goal in this section is to apply your learning to real life situations. You will
be given tasks which will demonstrate your understanding of the discriminant of
a quadratic equation.

DRAFT
March 24, 2014
76
b. How long will it take the ball to reach the height of 4.5 m?
How long will it take to touch the ground?
c. Do you think the ball can reach the height of 12 m? Why?
d. Will the ball hit the ring if the ring is 3 m high?
e. Write a similar situation but with varied initial height when the ball is thrown
with an initial upward velocity. Then model the path of the ball by a quadratic
expression.
f. Using the situation and the quadratic expression you have written in item e,
formulate and solve problems involving the height of the ball when it is
thrown after a given time.
2. Cite two more real-life situations where the discriminant of a quadratic equation is
being applied or illustrated.
SUMMARY/ SYNTHESIS/ GENERALIZATION
This lesson was about the nature of the roots of quadratic equations. The
lesson provided you with opportunities to describe the nature of the roots of
quadratic equations using the discriminant even without solving the equation. More
importantly, you were able to find out how the discriminant of a quadratic equation is
illustrated in real-life situations. Your understanding of this lesson and other
previously learned mathematical concepts and principles will facilitate your
understanding of the succeeding lessons.
In this section, your task was to solve problems involving the discriminant of a
quadratic equation. You were also asked to cite real-life situations where the
discriminant of a quadratic equation is applied or illustrated.
How did you find the performance task? How did the task help you see the real-

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March 24, 2014
77
LESSON NO. 4: THE SUM AND THE PRODUCT OF ROOTS OF
QUADRATIC EQUATIONS
What to KNOW:
Activity 1:Let’s do Addition and Multiplication!
Directions: Perform the indicated operation then answer the questions that follow.
1. 7 + 15 = 6. (8)(15) =
2. -9 + 14 = 7. (-4)(7) =
3. -6 + (-17) = 8. (-6)(-12)=
4. 




 
2
1
8
3
9. 










 
5
2
7
3
5. 




 





 
3
2
6
5
10. 




 





 
8
3
5
4
Questions:
a. How did you determine the result of each operation?
b. What mathematics concepts and principles did you apply to arrive at each result?
answers? If NOT, explain why.
mathematics concepts and principles previously studied and your skills in
performing mathematical operations. These knowledge and skills will help you
in understanding the sum and product of the roots of quadratic equations. As
you go through this lesson, think of this important question: “How do the sum
and product of roots of quadratic equation facilitate in understanding the
required conditions of real-life situations?” To find the answer, perform each
activity. If you find any difficulty in answering the exercises, seek the assistance
of your teacher or peers or refer to the modules you have gone over earlier.
You may check your answers with your teacher.

DRAFT
March 24, 2014
78
Activity 2: Find My Roots!
Directions: Find the roots of each of the following quadratic equations using any
method. Answer the questions that follow.
1. x² + 3x + 2 = 0 6. 15h² – 7h – 2 = 0
2. s² – 5s + 6 = 0 7. 12s² – 5s – 3 = 0
3. r² + 2r – 8 = 0 8. 6t² – 7t – 3 = 0
4. t² + 12t + 36 = 0 9. 3m² – 8m – 4 = 0
5. 4x² + 16x + 15 = 0 10. 2w² – 3w – 20 = 0
Questions:
a. How did you find the roots of each quadratic equation?
Which method of solving quadratic equations did you use in finding the roots?
b. Which quadratic equation did you find difficult to solve? Why?
answers? If NOT, explain why.
Activity 3: Relate Me to My Roots!
Directions: Use the quadratic equations below to answer the questions that follow.
You may work in groups of 4.
x2
+ 7x + 12 = 0 2x2
– 3x – 20 = 0
Were you able to perform each indicated operation correctly? In the next activity,
you will strengthen further your skills in finding the roots of quadratic equations.
Were you able to find the roots of each quadratic equation? In the next activity,
you will evaluate the sum and product of the roots and its relation to the
coefficients of the quadratic equation.

DRAFT
March 24, 2014
79
1. What are the values of a, b, and c in each equation?
a. x2 + 7x + 12 = 0; a = _____ b = _____ c = _____
b. 2x2 – 3x – 20 = 0; a = _____ b = _____ c = _____
2. Determine the roots of each quadratic equation using any method.
a. x2 + 7x + 12 = 0; x1 = _______ x2 = _______
b. 2x2 – 3x – 20 = 0; x1 = _______ x2 = _______
3. Complete the following table.
Quadratic Equation Sum of Roots Product of Roots
x2 + 7x + 12 = 0
2x2 – 3x – 20 = 0
4. What do you observe about the sum and the product of the roots of each
quadratic equation in relation to the values of a, b, and c?
5. Do you think a quadratic equation can be determined given its roots or solutions?
Justify your answer by giving 3 examples.
6. Do you think a quadratic equation can be determined given the sum and product
of its roots? Justify your answer by giving 3 examples.
Activity 4: What the Sum and Product Mean to Me…
A rectangular garden has an area of 132 m2 and a perimeter of 46 m.
Were you able to relate the values of a, b, and c of each quadratic equation with
the sum and product of its roots?

DRAFT
March 24, 2014
80
Questions:
1. What equation would describe the area of the garden? Write the equation in
terms of the width of the garden.
2. What can you say about the equation formulated in item 1?
3. Find the roots of the equation formulated in item 1. What do the roots represent?
4. What is the sum of the roots? How is this related to the perimeter?
5. What is the product of the roots? How is this related to area?
We now discuss how the sum and product of the roots of the quadratic
equation 0cba 2
 xx can be determined using the coefficients a, b, and c.
Remember that the roots of a quadratic equation can be determined using the
quadratic formula,
a2
ac4bb 2

x . From the quadratic formula, let
a2
ac4bb 2
1

x and
a2
ac4bb 2
2

x be the roots. Let us now find the sum
and the product of these roots.
Were you able to relate the sum and product of the roots of a quadratic equation
with its values of a, b, and c? Suppose you are asked to find the quadratic
equation given the sum and product of its roots, how will you do it? You will be
able to answer this as you perform the succeeding activities. However, before
performing these activities, read and understand first some important notes on
the sum and product of the roots of quadratic equations and the examples
presented.

DRAFT
March 24, 2014
81
Example 1: Find the sum and the product of the roots of 01082 2
 xx .
The values of a, b, and c in the equation are 2, 8, and -10, respectively.
Sum of the roots =
a
b

  4
2
8
a
b




Sum of the Roots of Quadratic Equation
a2
ac4bb
a2
ac4bb 22
21



 xx
a2
ac4bbac4bb 22
21

 xx
a2
b2
21

 xx 
a
b
21

 xx
The sum of the roots of quadratic equation is
a
b
.
Product of the Roots of Quadratic Equation







 







 

2a
4acbb
2a
4acbb 22
21 xx
 
 2
2
22
21
2a
4acbb 




 
 xx 
2
22
21
4a
4acbb 
 xx
221
4a
4ac
 xx 
a
c
21  xx
The product of the roots of quadratic equation is
a
c
.

DRAFT
March 24, 2014
82
The sum of the roots of 01082 2
 xx is 4 .
Product of the roots =
a
c
 5
2
10
a
c



The product of the roots of 01082 2
 xx is 5 .
To check, find the roots of 01082 2
 xx using any of the methods
of solving quadratic equations. Then determine the sum and the
product of the roots that will be obtained.
The roots of the equation are 1 and -5. Find the sum and the product of
these roots
Therefore, the sum and the product of the roots of 01082 2
 xx
are 4 and 5 , respectively.
Example 2: Use the values of a, b, and c in finding the roots of the quadratic
equation 01872
 xx .
The values of a, b, and c in the equation are 1, 7, and -18, respectively.
Use these values to find the sum and the product of the roots of the
equation.
Sum of the roots =
a
b

  7
1
7
a
b




The sum of the roots of 01872
 xx is 7 .
Product of the roots =
a
c
 18
1
18
a
c



The product of the roots of 01872
 xx is 18 .
Let 11 x and 52 x .
Sum of the roots:   45121  xx
Product of the roots:    55121  xx

DRAFT
March 24, 2014
83
If 1x and 2x are the roots of the quadratic equation 01872
 xx ,
then the sum and the product of its roots are as follows:
By inspection, the two numbers that give a sum of -7 and a product of
-18 are -9 and 2.
To check, let 91 x and 22 x then find their sum and product.
Therefore, the roots of the quadratic equation 01872
 xx are:
9x or 2x . These values of x make the equation true.
Sum of the roots: 721  xx
Product of the roots: 1821  xx
Sum: 721  xx
729
?

77 
721  xx is true for
91 x and 22 x .
Product: 1821  xx
   1829
?

1818 
1821  xx is true for
91 x and 22 x .
Learn more about
the Sum and the
Product of Roots of
Quadratic Equations
may open the
following links.
http://www.regentsprep.org/Regents/math/algtr
ig/ATE4/natureofroots.htm
http://www.youtube.com/watch?v=l7FI4T19uIA
http://www.athometuition.com/QuadraticEquati
onsFormula.php
http://www.math-help-ace.com/Quadratic-
Equation-Solver.html

DRAFT
March 24, 2014
84
What to PROCESS
Activity 5: This is My Sum and this is My Product. Who Am I?
Directions: Use the values of a, b, and c of each of the following quadratic equations
in determining the sum and the product of its roots. Verify your answers
by obtaining the roots of the equation. Answer the questions that follow.
1. x2 + 4x + 3 = 0 Sum: _____Product: _____ Roots: _____________
2. 6x2 + 12x – 18 = 0 Sum: _____ Product: _____ Roots: _____________
3. x2 + 4x – 21 = 0 Sum: _____ Product: _____ Roots: _____________
4. 2x2 + 3x – 2 = 0 Sum: _____ Product: _____ Roots: _____________
5. 3x2 – 10x – 8 = 0 Sum: _____ Product: _____ Roots: _____________
6. 4x2 + 8x + 3 = 0 Sum: _____ Product: _____ Roots: _____________
7. 9x2 – 6x = 8 Sum: _____ Product: _____ Roots: _____________
8. 8x2 = 6x + 9 Sum: _____ Product: _____ Roots: _____________
9. 10x2 – 19x + 6 = 0 Sum: _____ Product: _____ Roots: _____________
10. 2x2 – 3x = 0 Sum: _____ Product: _____ Roots: _____________
Questions:
a. How did you determine the sum and the product of the roots of each quadratic
equation?
Now that you learned about the sum and product of the roots of quadratic
equations, you may now try the activities in the next sections.
and principles in writing and in determining the roots of quadratic equations. Use
the mathematical ideas and the examples presented in the preceding section to

DRAFT
March 24, 2014
85
b. Do you think it is always convenient to use the values of a, b, and c of a quadratic
equation in determining its roots? Explain your answer.
c. What do you think is the significance of knowing the sum and the product of the
roots of quadratic equations?
Activity 6: Here Are The Roots. Where is the Trunk?
Directions: Write the quadratic equation in the form ax² + bx + c = 0 given the
following roots. Answer the questions that follow.
1. 5 and 9 6. -9 and 0
2. 8 and 10 7. 2.5 and 4.5
3. 6 and 3 8. -3 and -3
4. -8 and -10 9.
6
5
 and
6
1

5. -3 and 15 10.
3
2
 and
4
3
Questions:
a. How did you determine the quadratic equation given its roots?
b. What mathematics concepts or principles did you apply to arrive at the equation?
c. Are there other ways of getting the quadratic equation given the roots? If there
are any, explain and give examples.
d. Compare your answers with those of your classmates. Did you arrive at the same
Was it easy for you to determine the sum and the product of the roots of
quadratic equations? Were you able to find out the importance of knowing these
concepts? In the next activity, you will determine the quadratic equation given its
roots.
Were you able to determine the quadratic equation given its roots? Did you use
the sum and the product of the roots to determine the quadratic equation? I
know you did! Let us now find out how the sum and the product of roots are
illustrated in real life. Perform the next activity.

DRAFT
March 24, 2014
86
Activity 7: Fence my Lot!
Directions: Read and understand the situation below to answer the questions that
follow.
Mang Juan owns a rectangular lot. The perimeter of the lot is 90 m and its
area is 450 m2.
Questions:
1. What equation represents the perimeter of the lot? How about the equation that
represents its area?
2. How is the given situation related to the lesson, the sum and the product of roots
of quadratic equation?
3. Using your idea of the sum and product of roots of quadratic equation, how would
you determine the length and the width of the rectangular lot?
4. What are the dimensions of the rectangular lot?
In this section, the discussion was about the sum and product of the roots of
quadratic equation ax2 + bx + c = 0 and how they are related to the values of a,
b, and c.

DRAFT
March 24, 2014
87
Activity 8: Think of These Further!
1. The following are two different ways of determining a quadratic equation whose
roots are 5 and 12.
Method 1: x = 5 or x = 12  x – 5 = 0 or x – 12 = 0 Why?
(x – 5)(x – 12) = 0 Why?
Quadratic Equation: 060172
 xx Why?
Method 2: x1 = 5 or x2 = 12
Sum of the Roots: 21 xx  = 5 +12 = 17
21 xx  =
a
b
17

a
b
Why?
17
a
b
Why?
Product of the Roots:    6012521  xx
a
c
xx  21
60
a
c
Why?
Quadratic Equation: 0cba 2
 xx  02

a
c
x
a
b
x Why?
060172
 xx Why?
You goal in this section is to take a closer look at some aspects of the topic. You
are going to think deeper and test further your understanding of the sum and
product of roots of quadratic equations. After doing the following activities, you
should be able to answer this important question: “How do the sum and product
of roots of quadratic equation facilitate in understanding the required conditions
of real-life situations?”

DRAFT
March 24, 2014
88
a. Describe each method of finding the quadratic equation.
b. Which method of determining the quadratic equation do you think is easier to
follow? Why?
c. What do you think are the advantages and disadvantages of each method
used in determining the quadratic equation? Explain and give 3 examples.
2. Suppose the sum of the roots of a quadratic equation is given, do you think you
can determine the equation? Justify your answer.
3. The sum of the roots of a quadratic equation is -5. If one of the roots is 7, how
would you determine the equation? Write the equation.
4. Suppose the product of the roots of a quadratic equation is given, do you think
you can determine the equation? Justify your answer.
5. The product of the roots of a quadratic equation is 51. If one of the roots is -17,
what could be the equation?
6. The perimeter of a rectangular bulletin board is 20 ft. If the area of the board is
21 ft.2, what are its length and width?
What to TRANSFER
In this section, the discussion was about your understanding of the sum and
product of roots of quadratic equations.
What new insights do you have about the sum and product of roots of quadratic
equations? How would you connect this to real life? How would you use this in
making decisions?
be given a practical task in which you will demonstrate your understanding.

DRAFT
March 24, 2014
89
Activity 9: Let’s Make a Scrap Book!
Direction: Work in a group of 3 and make a scrap book that contains all the things
you have learned in this lesson. This includes the following:
1. A journal on how to determine a quadratic equation given the roots, or given the
sum and product of roots;
2. At least 5 examples of finding the quadratic equations given the roots, or given
the sum and product of the roots, and;
3. Three pictures showing the applications of sum and product of roots of quadratic
equations in real life. Describe how quadratic equations are illustrated in the
pictures.
SUMMARY/SYNTHESIS/GENERALIZATION:
This lesson was about the Sum and Product of Roots of Quadratic Equations.
In this lesson, you were able to relate the sum and product of the roots of quadratic
equation 0cba 2
 xx with its values of a, b, and c. Furthermore, this lesson has
given you an opportunity to find the quadratic equation given the roots. Your
understanding of this lesson and other previously learned mathematics concepts and
principles will facilitate your learning of the succeeding lessons.
In this section, your task was to make a journal on how to determine the
quadratic equation given its roots and to cite three real-life situations that
illustrate the applications of quadratic equations.
How did you find the performance tasks? How did the tasks help you see the
real-world use of the topic?

DRAFT
March 24, 2014
90
LESSON NO. 5: EQUATIONS TRANSFORMABLE TO QUADRATIC
EQUATIONS
What to KNOW:
Activity 1: Let’s Recall
Directions: Find the solution/s of the following quadratic equations. Answer the
1. x²– 4x + 4 = 0 4. 2m² + 5m + 2 = 0
2. s² – 3s – 10 = 0 5. 2n² + 2n -12 = 0
3. r2 + 5r – 14 = 0 6. 3p² + 7p + 4 = 0
Questions:
What method of solving quadratic equations did you use to find the roots of
each?
b. Compare your answers with those of your classmates. Did you arrive at the same
mathematics concepts and principles previously studied and your skills in
performing mathematical operations. These knowledge and skills will help you in
understanding the solution of equations that are transformable to quadratic
equations. As you go through this lesson, think of this important question: “How
does finding solutions of quadratic equations facilitate in solving real-life
problems?” To find the answer, perform each activity. If you find any difficulty in
answering the exercises, seek the assistance of your teacher or peers or refer to
the modules you have gone over earlier. You may check your answers with your
teacher.
Were you able to find the solution/s of the quadratic equations? In the next
activity, you will add or subtract rational algebraic expressions and express the
results in simplest forms. These mathematical skills are necessary for you to
solve equations that are transformable to quadratic equations.

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91
Activity 2: Let’s Add and Subtract!
Direction: Perform the indicated operation then express your answer in simplest
form. Answer the questions that follow.
1.
5
21 x
x
 4.
x
x
x
x
3
2
2
1 


2.
5
124 

x
x
5.
2
1
2
5




x
x
x
x
3.
x
xx 1
3
2 
 6.
2
2
1 

 xx
x
Questions:
a. How did you find the sum or the difference of the rational algebraic expressions?
b. What mathematics concepts or principles did you apply in adding or subtracting
the rational algebraic expressions?
c. How did you simplify the resulting expressions?
Activity 3: How Long Does It Take To Finish Your Job?
Directions: Read and understand the situation below, then answer the questions that
follow.
Were you able to add or subtract the rational expressions and simplify the
results? Suppose you were given a rational algebra equation, how would you
find its solution/s? You will learn this in the succeeding activities.
Mary and Carol are doing a math
project. Carol can do the work twice as
fast as Mary. If they work together, they
can finish the project in 4 hours. How
long does it take Mary working alone to
do the same project?

DRAFT
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92
Questions:
1. If Mary can finish the job in x hours alone, how many hours will it take Carol to do
the same job alone?
2. How would you represent the amount of work that Mary can finish in 1 hour? How
about the amount of work that Carol can finish in 1 hour?
3. If they work together, what equation would represent the amount of work they can
finish in 1 hour?
5. How would you solve the equation formulated? What mathematics concepts and
principles are you going to use?
There are equations that are transformable to quadratic equations. These
equations may be given in different forms. Hence, the procedures in transforming
these equations to quadratic equations may also be different.
Once the equations are transformed to quadratic equations, then they can be
solved using the techniques learned in previous lessons. The different methods of
solving quadratic equations, such as extracting square roots, factoring, completing
the square, and using the quadratic formula, can be used to solve these transformed
equations.
Solving Quadratic Equations That Are Not Written In Standard Form
Example 1: Solve x(x– 5) = 36.
This is a quadratic equation that is not written in standard form.
To write the quadratic equation in standard form, simplify the
expression x(x – 5).
How did you find the preceding activities? Are you ready to learn more about
rational algebraic equations? From the activities done, you were able to simplify
rational algebraic expressions. Also, you were able to represent quantities in
real-life using rational algebraic expressions and equations. But how are
quadratic equations used in solving real-life problems? You will find this out in
the activities in the next section. Before doing these activities, read and
understand first some important notes on equations that are transformable to
quadratic equations and the examples presented.

DRAFT
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93
x(x– 5) = 36  3652
 xx
Write the resulting quadratic equation in standard form.
3652
 xx  03652
 xx
Use any of the four methods of solving quadratic equations in finding
the solutions of the equation 03652
 xx .
Try factoring in finding the roots of the equation.
03652
 xx     049  xx
09 x or 04 x Why?
9x or 4x Why?
Check whether the obtained values of x make the equation
  365 xx true.
If the obtained values of x make the equation   365 xx true, then
the solutions of the equation are: 9x or 4x .
Example 2: Find the roots of the equation    3725 22
 xx .
The given equation is a quadratic equation but it is not written in
standard form. Transform this equation to standard form, then solve it
using any of the methods of solving quadratic equations.
    3725 22
 xx  37442510 22
 xxxx Why?
3742541022
 xxxx Why?
372962 2
 xx Why?
0862 2
 xx Why?
0862 2
 xx     0422  xx Why?
022 x or 04 x Why?
1x or 4x Why?

DRAFT
March 24, 2014
94
The solutions of the equation are: 1x or 4x . These values of x
make the equation     3725 22
 xx true.
Solving Rational Algebraic Equations Transformable to Quadratic Equations
Example 3: Solve the rational algebraic equation 2
4
36



x
x
.
The given rational algebraic equation can be transformed to a
quadratic equation. To solve the equation, the following procedure can
be followed.
a. Multiply both sides of the equation by the Least Common Multiple
(LCM) of all denominators. In the given equation, the LCM is 4x.
2
4
36



x
x
  24
4
36
4 x
x
x
x 




 

xxx 8324 2
 Why?
b. Write the resulting quadratic equation in standard form.
xxx 8324 2
  024112
 xx Why?
c. Find the roots of the resulting equation using any of the methods of
solving quadratic equations. Try factoring in finding the roots of the
equation.
024112
 xx     083  xx Why?
03 x or 08 x Why?
3x or 8x Why?
Check whether the obtained values of x make the equation
2
4
36



x
x
true.
If the obtained values of x make the equation 2
4
36



x
x
true, then
the solutions of the equation are: 3x or 8x .

DRAFT
March 24, 2014
95
Example 4: Find the roots of
2
4
1
2
8




x
x
x
x .
The equation
2
4
1
2
8




x
x
x
x is a rational algebraic equation that
can be written in the form 0cba 2
 xx .
To find the roots of the equation, you can follow the same procedure as
in the previous examples of solving rational algebraic equations.
a. Multiply both sides of the equation by the LCM of all denominators.
In the given equation, the LCM is x – 3.
2
4
1
2
8




x
x
x
x      














2
4
12
2
8
2
x
x
x
x
xx
2
42
2
822





x
xx
x
xx
Why?
xxxx 42822
 Why?
b. Write the resulting quadratic equation in standard form.
xxxx 42822
  25822
 xxx Why?
01072
 xx Why?
c. Find the roots of the resulting equation using any of the methods of
solving quadratic equations. Let us solve the equation by factoring.
01072
 xx     025  xx Why?
05 x or 02 x Why?
5x or 2x Why?
The equation 01072
 xx has two solutions, 5x or 2x . Check
whether the obtained values of x make the equation
2
4
1
2
8




x
x
x
x true.

DRAFT
March 24, 2014
96
The equation
2
4
1
2
8




x
x
x
x is true when 5x . Hence 5x is a
solution.
Observe that at 2x , the value of
2
8
x
is undefined or does not
exist. The same is true with
2
4
x
x
. Hence, 2x is an extraneous root
or solution of the equation
2
4
1
2
8




x
x
x
x . An extraneous root or
solution is a solution of an equation derived from an original equation.
However, it is not a solution of the original equation.
For 2x :
2
4
1
2
8




x
x
x
x 
 
22
24
1
22
8
2
?




0
8
1
0
8
2
?

For 5x :
2
4
1
2
8




x
x
x
x 
 
25
54
1
25
8
5
?




3
20
1
3
8
5
?

3
203
3
815 ? 


3
23
3
23


DRAFT
March 24, 2014
97
What to PROCESS:
Activity 4: View Me In Another Way!
Directions: Transform each of the following equations to a quadratic equation in the
form 0cba 2
 xx . Answer the questions that follow.
1.   25 xx 6. 10
4
5
5
2 2

xx
2.   156 2
s 7. 7
2
32

t
t
3.     932 22
 tt 8. 1
2
43
 x
xx
4.     10432 22
 rr 9. 3
2
5
5
6




s
s
5.     1574 22
 mm 10. 7
5
4
1
2



 rr
Questions:
a. How did you transform each equation into quadratic equation? What mathematics
concepts or principles did you apply?
b. Did you find any difficulty in transforming each equation into a quadratic
equation? Explain.
Learn more about
Equations Transformable
to Quadratic Equations
links.
http://www.analyzemath.com/Algebra2/Algeb
ra2.html
http://mathvids.com/lesson/mathhelp/1437-
solving-rational-equation-to-quadratic-1
http://www.analyzemath.com/Algebra2/algeb
ra2_solutions.html
Your goal in this section is to transform equations into quadratic equations and
solve these. Use the mathematical ideas and examples presented in the
preceding section to answer the activities provided.

DRAFT
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98
answer? If NOT, explain.
Activity 5: What Must Be The Right Value?
Directions: Find the solution/s of each of the following equations. Answer the
Questions:
a. How did you solve each equation? What mathematics concepts or principles did
you apply to solve each equation?
b. Which equation did you find difficult to solve? Why?
d. Do you think there are other ways of solving each equation? Show these if there
are any.
Were you able to solve the given equations? Was it easy for you to transform
those equations into quadratic equations? In the next activity, you will solve
equations that are transformable to quadratic equations.
Eq.1
Eq.3
Eq.2
Were you able to transform each equation into a quadratic equation? Why do
you think there is a need for you to do such activity? Find this out in the next
activity.
1
6
4
3
1

x
x
    1531 22
 xx
  2110 xx

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99
Activity 6: Let’s Be True!
Directions: Find the solution set of the following.
1.   283 xx 6.
3
2
6
1

x
x
2.   sss 1223  7. 2
23
4


t
t
3.     4581 22
 tt 8. 1
3
2
4
5


 x
x
x
4.     65213 22
 rr 9.
2
1
4
1
2
2 



 s
s
s
5.
   
3
16
3
2
5
2 22



 xx
10. 3
3
1
5
2



 xx
x
Activity 7: Let’s Paint the House!
Directions: Read and understand the situation below, then answer the questions that
follow.
Jessie and Mark are planning to paint a house together. Jessie thinks that if
he works alone, it would take him 5 hours more than the time Mark takes to paint the
entire house. Working together, they can complete the job in 6 hours.
Questions:
1. If Mark can finish the job in m hours, how long will it take Jessie to finish the job?
2. How would you represent the amount of work that Mark can finish in 1 hour? How
about the amount of work that Jessie can finish in 1 hour?
3. If they work together, what equation would represent the amount of work they can
finish in 1 hour?
5. How will you solve the equation formulated? What mathematics concepts and
principles are you going to use?
Were you able to find the solution of each equation above? Now it’s time to
apply those equations in solving real-life problems. In the next activity, you will
solve real-life problems using your knowledge in solving rational algebraic
equations.

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100
Activity 8: My Understanding of Equations Transformable to Quadratic
Equations
1. How do you transform a rational algebraic equation into a quadratic equation?
Explain and give examples.
2. How do you determine the solutions of quadratic equations? How about rational
algebraic equations transformable to quadratic equations?
3. Suppose a quadratic equation is derived from a rational algebraic equation. How
do you check if the solutions of the quadratic equation are also the solutions of
the rational algebraic equation?
4. Which of the following equations have extraneous roots or solutions? Justify your
answer.
a.
12
7
1
11



xx
c. 2
8
63 2



x
x
x
b. x
x
xx
215
5
52



d.
5
8
3
2
5
43




x
x
In this section, the discussion was about the solutions of equations
transformable to quadratic equations.
You goal in this section is to take a closer look at some aspects of the topic. You
are going to think deeper and test further your understanding of the solution of
equations that are transformable to quadratic equations. After doing the
following activities, you should be able to answer this important question: How
does the concept of quadratic equation used in solving real-life problems?

DRAFT
March 24, 2014
101
5. In a water refilling station, the time that a pipe takes to fill a tank is 10 minutes
more than the time that another pipe takes to fill the same tank. If the two pipes
are opened at the same time, they can fill the tank in 12 minutes. How many
minutes does each pipe take to fill the tank?
What to TRANSFER:
Activity 9: A Reality of Rational Algebraic Equation
Directions: Cite a real-life situation where the concept of a rational algebraic equation
transformable to a quadratic equation is being applied. Use the situation to answer
the following questions.
1. How is the concept of a rational algebraic equation transformable to a quadratic
equation applied in the situation?
2. What quantities are involved in the situation? Which of these quantities are
known? How about the quantities that are unknown?
3. Formulate, then solve a problem out of the given situation.
4. What do the solutions obtained represent? Explain your answer.
In this section, the discussion was about your understanding of equations
transformable to quadratic equations. What new insights do you have about this
lesson? How would you connect this to real life? How would you use this in
making decisions?
be given tasks which will demonstrate your understanding.

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102
Rubric: Real-Life Situations Involving Rational Algebraic Equations
Transformable to Quadratic Equations
4 3 2 1
The situation is
clear, realistic and
the use of a
rational algebraic
equation
transformable to a
quadratic equation
and other
mathematics
concepts are
properly illustrated.
The situation is
clear but the use of
a rational algebraic
equation
transformable to a
quadratic equation
and other
mathematic
concepts are not
The situation is not
so clear, and the
use of a rational
algebraic equation
transformable to a
quadratic equation
is not illustrated.
The situation is
not clear and the
use of a rational
algebraic
equation
transformable to
a quadratic
equation is not
illustrated.
Score Descriptors
6
Poses a more complex problem with 2 or more correct
possible solutions and communicates ideas unmistakably,
shows in-depth comprehension of the pertinent concepts
and/or processes and provides explanations wherever
appropriate.
5
Poses a more complex problem and finishes all significant
parts of the solution and communicates ideas unmistakably,
and/or processes.
4
Poses a complex problem and finishes all significant parts of
depth comprehension of the pertinent concepts and/or
processes.
3
Poses a complex problem and finishes most significant parts
of the solution and communicates ideas unmistakably, shows
2
Poses a problem and finishes some significant parts of the
solution and communicates ideas unmistakably but shows
gaps on theoretical comprehension.
1
Source: D.O. #73 s. 2012

DRAFT
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103
This lesson was about the solutions of equations that are transformable to
quadratic equations including rational algebraic equations. This lesson provided you
with opportunities to transform equations into the form 02
 cbxax and to solve
these. Moreover, this lesson provided you with opportunities to solve real-life
problems involving rational algebraic equations transformable to quadratic equations.
Your understanding of this lesson and other previously learned mathematics
concepts and principles will facilitate your understanding of the succeeding lessons.
In this section, your task was to cite a real-life situation where the concept of a
rational algebraic equation transformable to a quadratic equation is illustrated.
How did you find the performance tasks? How did the tasks help you see the
real-world use of the topic?

DRAFT
March 24, 2014
104
LESSON NO. 6: SOLVING PROBLEMS INVOLVING QUADRATIC
EQUATIONS
What to KNOW:
Activity 1: Find My Solutions!
Directions: Solve each of the following quadratic equations. Explain how you arrived
at your answers.
1. x(2x – 5) = 0 6. (3m + 4)(m – 5) = 0
2. 2t(t – 8) = 0 7. k2 – 4k – 45 = 0
3. 6x(2x + 1) = 0 8. 2t2 – 7t – 49 = 0
4. (r + 2)(r + 13) = 0 9. 3w2 – 11w = 4
5. (h – 4)(h – 10) = 0 10. 4u2 + 4u = 15
mathematical operations. These knowledge and skills may help you in
understanding the solutions to real-life problems involving quadratic equations.
As you go through this lesson, think of this important question: How are
quadratic equations used in solving real-life problems and in making decisions?”
To find the answer, perform each activity. If you find any difficulty in answering
the exercises, seek the assistance of your teacher or peers or refer to the
modules you have gone over earlier. You may check your answers with your
teacher.
Were you able to find the solution of each quadratic equation? In the next
activity, you will translate verbal phrases into mathematical expressions. This
will help you solve real-life problems later on.

DRAFT
March 24, 2014
105
Activity 2: Translate into…
Directions: Use a variable to represent the unknown quantity then write an equation
from the given information. Explain how you arrived at your answer.
1. The area of the concrete pathway is 350 m2 and its perimeter pathway is 90 m.
What is the length of the pathway?
2. A rectangular lot has an area of 240 m2. What is the width of the lot if it requires
64 m of fencing materials to enclose it?
3. The area of a garden is 160 m2. Suppose the length of the garden is 3 m more
than twice its width. What is the length of the garden?
4. The length of a tarpaulin is 3 ft. more than thrice its width and its area is 126 ft.².
What is the length of the tarpaulin?
5. Mario and Kenneth work in a car wash station. The time that Mario takes in
washing a car alone is 20 minutes less than the time that Kenneth takes in
washing the same car. If both of them work together in washing the car, it will
take them 90 minutes. How long will it take each of them to wash the car?
Activity 3: What are My Dimensions?
The length of a rectangular floor is 5 m longer than its width. The area of the
floor is 84 m2.
Questions:
1. What expression represents the width of the floor? How about the expression that
represents its length?
Were you able to represent each situation by an equation? If YES, then you are
ready to perform the next activity.

DRAFT
March 24, 2014
106
2. Formulate an equation relating the width, length and the area of the floor. Explain
how you arrived at the mathematical sentence.
3. How would you describe the equation that you formulated?
4. Using the equation, how will you determine the length and width of the floor?
5. What is the width of the floor? How about its length?
6. How did you find the length and width of the floor?
The concept of quadratic equations is illustrated in many real-life situations.
Problems that arise from these situations, such as those involving area, work, profits,
and many others, can be solved by applying the different mathematics concepts and
principles previously studied including quadratic equations and the different ways of
solving them.
Example 1: A rectangular table has an area of 27 ft.2 and a perimeter of 24 ft. What
are the dimensions of the table?
The product of the length and width of the rectangular table represents
its area. Hence, length (l) times width (w) = 27 or lw = 27.
Also, twice the sum of the length and the width of the table gives the
perimeter. Hence, 2l + 2w = 24.
If we divide both sides of the equation 2422  wl by 2, then
12 wl .
We can think of lw = 27 and 12 wl as the equations representing
the product and sum of roots, respectively, of a quadratic equation.
Remember that if the sum and the product of the roots of a quadratic
equation are given, the roots can be determined. This can be done by
inspection or by using the equation 02

a
c
x
a
b
x , where
a
b
 is the
sum of the roots and
a
c
is the product.
Were you able to find the width and length of the rectangular floor correctly? In
the next activity, you will find out how to solve real-life problems using quadratic
equations but before we proceed to the next activities, read and understand the
following note in solving word and real-life problems.

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March 24, 2014
107
By inspection, the numbers whose product is 27 and whose sum is 12
are 3 and 9.
Product: 2793 
Sum: 1293 
The roots of the quadratic equation then are 3 and 9. This implies that
the width of the table is 3 ft. and its length is 9 ft.
Another method of finding the roots is to use the equation
02

a
c
x
a
b
x . Let 12
a
b
or 12
a
b
and 27
a
c
. Then substitute
these values in the equation.
02

a
c
x
a
b
x    027122
 xx
 027122
 xx
Solve the resulting equation 027122
 xx using any of the
methods of solving quadratic equation. Try factoring.
027122
 xx     093  xx
 03 x or 09 x
 3x or 9x
With the obtained roots of the quadratic equation, the dimensions of
the table then are 3 ft. and 9 ft., respectively.
Example 2: An amusement park wants to place a new rectangular billboard to
inform visitors of their new attractions. Suppose the length of the
billboard to be placed is 4 m longer than its width and the area is 96
m². What will be the length and the width of the billboard?
If we represent the width, in meters, of the billboard by x, then its length
is x + 4. Since the area of the billboard is 96 m², then    964 xx .
The equation    964 xx is a quadratic equation that can be written
in the form 0cba 2
 xx .

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108
   964 xx  9642
 xx
09642
 xx
Solve the resulting equation.
09642
 xx     0128  xx
08 x or 012 x
8x or 12x
The equation 09642
 xx has two solutions: x = 8 or x = -12.
However, we only consider the positive value of x since the situation
involves measure of length. Hence, the width of the billboard is 8 m
and its length is 12 m.
Learn more about
the Applications of
Quadratic Equations
links.
http://www.mathsisfun.com/algebra/quadratic-
equation-real-world.html
http://www.purplemath.com/modules/quadprob
.htm
http://tutorial.math.lamar.edu/Classes/Alg/Qua
draticApps.aspx
http://www.algebra.com/algebra/homework/qu
adratic/lessons/Using-quadratic-equations-to-
solve-word-problems.lesson
http://www.slideshare.net/jchartiersjsd/quadrati
c-equation-word-problems
http://www.tulyn.com/algebra/quadratic-
equations/wordproblems
http://www.uwlax.edu/faculty/hasenbank/archiv
ed/mth126fa08/notes/11.10%20-
%20Quadratic%20Applications.pdf
http://www.pindling.org/Math/CA/By_Examples
/1_4_Appls_Quadratic/1_4_Appls_Quadratic.h
tml

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109
What to PROCESS:
Activity 4: Let Me Try!
Directions: Answer each of the following.
1. A projectile that is fired vertically into the air with an initial velocity of 120 ft. per
second can be modeled by the equation s = 120t – 16t2. In the equation, s is the
distance in feet of the projectile above the ground after t seconds.
a. How long will it take for a projectile to reach 216 feet?
b. Is it possible for the projectile to reach 900 feet? Justify your answer.
2. The length of a rectangular parking lot is 36 m longer than its width. The area of
the parking lot is 5,152 m2.
a. How would you represent the width of the parking lot? How about its length?
b. What equation represents the area of the parking lot?
c. How would you use the equation representing the area of the parking lot in
finding its length and width?
d. What is the length of the parking lot? How about its width? Explain how you
arrived at your answer.
e. Suppose the area of the parking lot is doubled, would its length and width also
double? Justify your answer.
3. The perimeter of a rectangular swimming pool is 86 m and its area is 450 m2.
Now that you have learned how to solve real-life problems involving quadratic
equations, you may now try the activities in the next sections.
Your goal in this section is to apply the key concepts of quadratic equations in
solving real-life problems. Use the mathematical ideas and the examples
presented in the preceding sections to answer the succeeding activities.

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110
a. How would you represent the length and the width of the swimming pool?
b. What equation represents the perimeter of the swimming pool?
How about the equation that represents its area?
c. How would you find the length and the width of the swimming pool?
d. What is the length of the swimming pool? How about its width? Explain how
you arrived at your answer.
e. How would you check if the dimensions of the swimming pool obtained satisfy
the conditions of the given situation?
f. Suppose the dimensions of the swimming pool are both doubled, how would it
affect its perimeter? How about its area?
In this section, the discussion was about solving real-life problems involving
quadratic equations.
Now that you know the important ideas about quadratic equations and their
applications in real life, let’s go deeper by moving on to the next section.
Your goal in this section is to take a closer look at some aspects of the topic.
You are going to think deeper and test further your understanding of the real-life
applications of quadratic equations. After doing the following activities, you
should be able to answer this important question: How are quadratic equations
used in solving real life problems and in making decisions?”

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111
Activity 5: Find Those Missing!
Directions: Solve the following problems. Explain how you arrived at your answers.
1. A rectangular garden has an area of 84 m2 and a perimeter of 38 m. Find its
length and width.
2. A children’s park is 350 m long and 200 m wide. It is surrounded by a pathway of
uniform width. Suppose the total area of the park and the pathway is 74,464 m2.
How wide is the pathway?
3. A car travels 20 kph faster than a truck. The car covers 350 km in two hours less
than the time it takes the truck to travel the same distance. What is the speed of
the car? How about the truck?
4. Jane and Maria can clean the house in 8 hours if they work together. The time
that Jane takes in cleaning the house alone is 4 hours more than the time Maria
takes in cleaning the same house. How long does it take Jane to clean the house
alone? How about Maria?
5. If an amount of money P in pesos is invested at r percent compounded annually,
it will grow to an amount A = P(1 + r)2 in two years. Suppose Miss Madrigal wants
her money amounting to Php200,000 to grow to Php228,980 in two years. At
what rate must she invest her money?
What to TRANSFER:
equations and their real-life applications.
What new insights do you have about the real-life applications of quadratic
equations? How would you connect this to your daily life? How would you use
be given tasks which will demonstrate your understanding of the lesson.

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112
Activity 6: Let’s Draw!
Directions: Make a design or sketch plan of a table than can be made out of ¾” x 4’ x
8’ plywood and 2” x 3” x 8’ wood. Using the design or sketch plan,
formulate problems that involve quadratic equations, then solve in as
many ways as possible.
4 3 2 1
The sketch plan is
accurately made,
presentable, and
appropriate.
The sketch plan is
accurately made
and appropriate.
The sketch plan is
not accurately
made but
appropriate.
The sketch plan
is made but not
appropriate.
Quadratic
equations are
accurately
formulated and
solved correctly.
Quadratic
equations are
accurately
formulated but not
all are solved
correctly.
Quadratic
equations are
accurately
formulated but are
not solved
correctly.
Quadratic
equations are
accurately
formulated but
are not solved.
Activity 7: Play the Role of…
Directions: Cite and role play a situation in real life where the concept of the
quadratic equation is applied. Formulate and solve problems out of these situations.
Rubric for Real-Life Situation Involving Quadratic Equation
4 3 2 1
The situation is
clear, realistic and
the use of the
quadratic equation
and other
mathematical
concepts are
properly
illustrated.
The situation is
clear but the use of
the quadratic
equation and other
mathematic
concepts are not
The situation is not
so clear, and the
use of the
quadratic equation
is not illustrated.
The situation is
not clear and the
use of the
quadratic equation
is not illustrated.

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113
Score Descriptors
6
Poses a more complex problem with 2 or more correct
possible solutions and communicates ideas unmistakably,
and/or processes and provides explanations wherever
appropriate.
5
Poses a more complex problem and finishes all significant
parts of the solution and communicates ideas unmistakably,
and/or processes.
4
Poses a complex problem and finishes all significant parts of
depth comprehension of the pertinent concepts and/or
processes.
3
Poses a complex problem and finishes most significant parts
of the solution and communicates ideas unmistakably, shows
2
Poses a problem and finishes some significant parts of the
solution and communicates ideas unmistakably but shows
gaps on theoretical comprehension.
1
Source: D.O. #73 s. 2012
This lesson was about solving real-life problems involving quadratic
equations. The lesson provided you with opportunities to see the real-life
applications of quadratic equations. Moreover, you were given opportunities to
formulate and solve quadratic equations based on real-life situations. Your
understanding of this lesson and other previously learned mathematics concepts and
principles will facilitate your understanding of the succeeding lessons.

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114
LESSON NO. 7: QUADRATIC INEQUALITIES
What to KNOW:
Activity 1: What Makes Me True?
Directions: Find the solution/s of each of the following mathematical sentences.
1. 85 x 6. 0652
 xx
2. 103 r 7. 0782
 tt
3. 2172 s 8. 1872
 rr
4. 1323 t 9. 01252 2
 hh
5. 8512  m 10. 49 2
s
Questions:
a. How did you find the solution/s of each mathematical sentence?
solution/s?
c. Which mathematical sentence has only one solution? More than one solution?
Describe these mathematical sentences.
Start Lesson 7 of this module by assessing your knowledge of the different
understanding quadratic inequalities. As you go through this lesson, think of this
important question: “How are quadratic inequalities used in solving real-life
problems and in making decisions?” To find the answer, perform each activity. If
you find any difficulty in answering the exercises, seek the assistance of your
teacher or peers or refer to the modules you have gone over earlier. You may
check your work with your teacher.
How did you find the activity? Were you able to find the solution/s of each
mathematical sentence? Did you find difficulty in solving each mathematical
sentence? If not, then you are ready to proceed to the next activity.

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115
Activity 2: Which Are Not Quadratic Equations?
Directions: Use the mathematical sentences below to answer the questions that
follow.
1. Which of the given mathematical sentences are quadratic equations?
2. How do you describe quadratic equations?
3. Which of the given mathematical sentences are not quadratic equations? Why?
4. How would you describe those mathematical sentences which are not quadratic
equations?
How are they different from those equations which are quadratic?
Activity 3: Let’s Do Gardening!
1. Illustrate the different rectangular gardens that Mr. Bayani could make.
2. What are the dimensions of the different gardens that Mr. Bayani wants to make?
3. What is the area of each garden in item 2?
4. What is the area of the smallest garden that Mr. Bayani can make? How about
the area of the largest garden? Explain your answer.
Mr. Bayani has a vacant lot in his
backyard. He wants to make as many
rectangular gardens as possible such that the
length of each garden is 2 m longer than its
width. He also wants the length of the garden
of smallest area to be 3 m.
02092
 xx
0572 2
 ss 0254 2
x 762
 mm
16102
 rrtt 9212 2
 0123 2
 ww
10615 2
 h
In the activity done, were you able to distinguish mathematical sentences which
are quadratic equations and which are not quadratic equations? Were you able
to describe mathematical sentences that make use of equality and inequality
symbols? In the next activity, you will learn how mathematical sentences
involving inequalities are illustrated in real life.

DRAFT
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116
5. What general mathematical sentence would represent the possible areas of the
gardens? Describe the sentence.
6. Using the mathematical sentence formulated, do you think you can find other
possible dimensions of the gardens that Mr. Bayani wants to make? If YES, how?
If NOT, explain.
7. Suppose the length of each garden that Mr. Bayani wants to make is 3 m longer
than its width and the area of the smallest garden is 10 m2. What general
mathematical sentence would represent the possible areas of the gardens? How
are you going to solve the mathematical sentence formulated? Find at least 3
possible solutions of the mathematical sentence.
8. Draw a graph to represent the solution set of the mathematical sentence
formulated in item 7. What does the graph tell you?
9. Are all solutions that can be obtained from the graph true to the given situation?
Why?
A quadratic inequality is an inequality that contains a polynomial of degree 2
and can be written in any of the following forms.
Examples: 1. 0152 2
 xx 3. 053- 2
 rr
2. ss 292
 4. 1042
 tt
To solve a quadratic inequality, find the roots of its corresponding equality.
The points corresponding to the roots of the equality, when plotted on the number
How did you find the preceding activities? Do you think you are already equipped
with those knowledge and skills needed to learn the new lesson? I’m sure you
are!!! From the activities done, you were able to find the solutions of different
mathematical sentences. You were able to differentiate quadratic equations from
those which are not. More important, you were able to perform an activity that will
lead you in understanding the new lesson. But how are quadratic inequalities
used in solving real-life problems and in making decisions? You will find these
out in the activities in the next section. Before doing these activities, read and
understand first some important notes on Quadratic Inequalities and the
examples presented.
0cba 2
 xx 0cba 2
 xx
0cba 2
 xx 0cba 2
 xx
where a, b, and c are real numbers and a ≠ 0.

DRAFT
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117
line, separates the line into two or three intervals. An interval is part of the solution of
the inequality if a number in that interval makes the inequality true.
Example 1: Find the solution set of 01272
 xx .
The corresponding equality of 01272
 xx is 01272
 xx .
Solve 01272
 xx .
   043  xx Why?
03 x and 04 x Why?
3x and 4x Why?
Plot the points corresponding to -3 and -4 on the number line.
The three intervals are: 4 x , 34  x , and  x3 .
Test a number from each interval against the inequality.
For 4 x ,
let x = -7.
For 34  x ,
let x = -3.6
For  x3 ,
let x = 0
01272
 xx
    012777
?
2

0124949
?

012  (True)
01272
 xx
    0126.376.3
?
2

0122.2596.12
?

024.0  (False)
01272
 xx
    012070 2
?

101200
?

1012  (True)
We also test whether the points x = –3 and x = –4 satisfy the inequality.
01272
 xx
    012373 2
?

012219
?

00  (False)
01272
 xx
    012474 2
?

0122816
?

00  (False)

DRAFT
March 24, 2014
118
Therefore, the inequality is true for any value of x in the interval
4 x or  x3 , and these intervals exclude –3 and –4.
The solution set of the inequality is 3or4:  xxx , and its graph
is shown below.
Note that hollow circles are used in the graph to show that –3 and –4
are not part of the solution set.
Another way of solving the quadratic inequality 01272
 xx is by following
the procedure in solving quadratic equations. However, there are cases to be
considered. Study the procedure in solving the quadratic inequality 01272
 xx
below. Discuss the reason for each step followed.
   043  xx Why?
Case 1:
    04and03  xx
Why?
Case 2:
    04and03  xx
Why?
The solution set of the inequality is  3or4:  xxx . Why?
To check, consider any number greater than -3 or less than -4. Substitute this
number to x in the inequality 01272
 xx .
Consider -2 and 3 which are both greater than -3.
When x = -2:
01272
 xx      012272
?
2

012144
?

02  (True)
Case 1
    04and03  xx
Case 2
    04and03  xx Why?
x + 3 > 0 and x + 4 > 0
x > –3 and x > – 4
x + 3 < 0 and x + 4 < 0
x < –3 and x < – 4
x > –3 x < – 4
Notice that the quadratic expression 1272
 xx is greater than zero or
positive. If we write the expression in factored form, what must be true about its
factors?

DRAFT
March 24, 2014
119
When x = 3:
01272
 xx      012373
?
2

012219
?

042  (True)
This shows that 01272
 xx is true for values of x greater than -3.
This time, consider -5 and -8 which are both less than -4.
When x = -5:
01272
 xx      012575
?
2

0123525
?

02  (True)
When x = -8:
01272
 xx      012878
?
2

0125664
?

020  (True)
The inequality 01272
 xx is also true for values of x less than -4.
When x = -3:
01272
 xx      012373
?
2

012219
?

00
?
 (Not True)
The inequality is not true for x = -3.
When x = -3.5:
01272
 xx      0125.375.3
?
2

0125.2425.12
?

025.0
?
 (Not True)
The inequality is not true for x = -3.5.
Will the inequality be true for any value of x greater than or equal to -4 but less
than or equal to -3?

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March 24, 2014
120
This shows that 01272
 xx is not true for values of x greater than
or equal to -4 but less than or equal to -3.
Example 2: 352 2
 xx
Rewrite 352 2
 xx to 0352 2
 xx . Why?
   0312  xx Why?
Case 1:
    03and012  xx
Why?
Case 2:
    03and012  xx
Why?
The solution set of the inequality is






 3
2
1
: xx . Why?
The figure below shows the graph of the solution set of the
inequality.
Note that
2
1
 and 3 are represented by points, to indicate that
they are part of the solution set.
Case 1
    03and012  xx
Case 2
    03and012  xx Why?
2x + 1  0 and x – 3  0
2x  –1 and x  3
2
1
x and 3x
2x + 1  0 and x – 3  0
2x  –1 and x  3
2
1
x and 3x
No solution
3
2
1
 x
Notice that the quadratic expression 352 2
 xx is less than or equal to zero.
If we write the expression in factored form, the product of these factors must
be zero or negative to satisfy the inequality. Remember that if the product of
two numbers is zero, either one or both factors are zeros. Likewise, if the
product of two numbers is negative, then one of these numbers is positive and
the other is negative.

DRAFT
March 24, 2014
121
To check, consider any number greater than or equal to
2
1
 but
less than or equal to 3. Substitute this number to x in the
inequality 352 2
 xx .
When x = 0:
352 2
 xx     
?
2
30502 
300
?

30  (True)
When x = 2:
352 2
 xx     
?
2
32522 
3108
?

32  (True)
This shows that 352 2
 xx is true for values of x greater than
or equal to
2
1
 but less than or equal to 3.
When x = -2:
352 2
 xx     
?
2
32522 
3108
?

318
?
 (Not True)
When x = 5:
352 2
 xx     
?
2
35552 
32550
?

325
?
 (Not True)
This shows that 352 2
 xx is not true for values of x less than
2
1

or greater than 3.
There are quadratic inequalities that involve two variables. These inequalities
can be written in any of the following forms.
Will the inequality be true for any value of x less than
2
1
 or greater than 3?

DRAFT
March 24, 2014
122
Examples: 1. 232
 xxy 3. 2
-49 xy 
2. 152 2
 xxy 4. 2
27 xxy 
How are quadratic inequalities in two variables illustrated in real life?
The following is a situation where quadratic inequality in two variables is
illustrated.
In the given situation, the width of the room can be represented by w and the
length by w + 15. Why?
If we represent the area of the old playground as A, then the quadratic
inequality that would represent the given situation is   Aww 15 or Aww 152
.
Why?
The situation tells us that the area of the new playground is greater than the
area of the old playground. This means that the area of the new playground is
greater than 2,200. It could be 2,300, 3,500, 4,600, and so on.
One possible pair of dimensions of the new playground is 50 m and 65 m,
respectively. With these dimensions, the area of the new playground is   m65m50
or 3,250 m2.
It is not possible for w to take a negative value because the situation involves
measures of length.
The solution set of quadratic inequalities in two variables can be determined
graphically. To do this, write the inequality as an equation, then show the graph. The
graph of the resulting parabola will be used to graph the inequality.
cbay 2
 xx cbay 2
 xx
cbay 2
 xx cbay 2
 xx
where a, b, and c are real numbers and a ≠ 0.
The city government is planning to construct a new children’s playground. It
wants to fence in a rectangular ground using one of the walls of a building.
The length of the new playground is 15 m longer than it is wide and its area is
greater than the old playground.
Suppose the area of the old playground is 2,200 m2. What could be the area
of the new playground? What could be its length and width? Is it possible that
the value of w is negative? Why?

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123
Example 1: Find the solution set of 232
 xxy .
Write the inequality to its corresponding equation.
232
 xxy  232
 xxy
Construct table of values for x and y.
Use these points to graph a parabola. Points B(0,8), C(1,6), and
E(-3,2) are points along the parabola. The coordinates of these
points do not satisfy the inequality 232
 xxy . Therefore,
they are not part of the solution set of the inequality. We use a
broken line to represent the parabola since the points on the
parabola do not satisfy the given inequality.
The parabola partitions the plane into two regions. Select one
point in each region and check whether the given inequality is
satisfied. For example, consider the origin (0, 0), and substitute
this in the inequality. We obtain 0 < 0 + 0 + 2 or 0 < 2, which is
correct. Therefore, the entire region containing (0, 0) represents
the solution set and we shade it. On the other hand, the point (0,
8) is on the other region. If we substitute this in the inequality,
we obtain 8 < 0 + 0 + 2 or 8 < 2, which is false. Therefore, this
x -5 -4 -3 -2 -1 0 1 2
y 12 6 2 0 0 2 6 12
Parabola
A(-6,7)
B(0,8)
C(1,6)
D(3,3)
E(-3,2)
F(-2,-3)
y
x

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124
region is not part of the solution set and we do not shade this
region.
To check, points A(-6,7), D(3,3), and F(-2,-3) are some of the
points in the shaded region. If the coordinates of these points
are substituted in 232
 xxy , the inequality becomes true.
Hence, they are part of the solution set.
Example 2: Find the solution set of 132 2
 xxy .
The figure in the preceding page shows the graph of
132 2
 xxy . All points in the shaded region including those
along the solid line (parabola) make up the solution set of the
inequality. The coordinates of any point in this region make the
inequality true. Points B(1,3), C(3,10), D(0,6), and E(0,1) are
some of the points on the shaded region and along the
parabola. The coordinates of these points satisfy the inequality
132 2
 xxy .
Consider point B whose coordinates are (1,3). If x = 1 and y = 3
are substituted in the inequality, then the mathematical
statement becomes true.
Parabola
y
x
A(-4,6)
B(1,3)
C(3,10
)
D(0,6)
E(0,1)
A(6,6)
F(-3,-1)
G(4,-2)

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125
132 2
 xxy      113123 2
?

1323
?

03 
Hence, (1,3) is a solution to the inequality.
What to PROCESS:
Activity 4: Quadratic Inequalities or Not?
Directions: Determine whether each mathematical sentence is a quadratic inequality
or not. Answer the questions that follow.
1. 01492
 xx 6. 0203 m
2. 153 2
 ss 7. 0)4)(52(  rr
Your goal in this section is to apply the key concepts of quadratic inequalities.
Use the mathematical ideas and the examples presented in the preceding
section to answer the activities provided.
Learn more about
links.
http://www.mathwarehouse.com/quadratic-
inequality/how-to-solve-and-graph-
quadratic-inequality.php
http://www.regentsprep.org/regents/math/alg
trig/ate6/quadinequal.htm
http://www.mathsisfun.com/algebra/inequalit
y-quadratic-solving.html
http://www.wtamu.edu/academic/anns/mps/
math/mathlab/col_algebra/col_alg_tut23_qu
adineq.htm
http://www.shelovesmath.com/algebra/inter
mediate-algebra/quadratic-inequalities/
http://www.cliffsnotes.com/math/algebra/alg
ebra-ii/quadratics-in-one-variable/solving-
quadratic-inequalities

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126
3. 0274 2
 tt 8. 112
 xx
4. 3102
 xx 9.     03294 2
 hh
5. 0512 2
 xx 10. 2
3215 xx 
Questions:
a. How do you describe quadratic inequalities?
b. How are quadratic inequalities different from linear inequalities?
c. Give at least three examples of quadratic inequalities.
Activity 5: Describe My Solutions!
Directions: Find the solution set of each of the following quadratic inequalities then
graph. Answer the questions that follow.
1. 01492
 xx 6. 012112 2
 tt
2. 016102
 rr 7. 0583 2
 xx
3. 562
 xx 8. 14 2
p
4. 1072
 mm 9. 01432 2
 xx
5. 01452
 xx 10. 523 2
 qq
Questions:
a. How did you find the solution set of each inequality?
solution set of each mathematical sentence?
c. How did you graph the solution set of each inequality?
d. How would you describe the graph of the solution set of a quadratic inequality?
e. How many solutions does each inequality have?
f. Are the solution/s of each inequality real numbers? Why?
Were you able to determine which mathematical sentences are quadratic
inequalities? In the next activity, you will find and describe the solution sets of
quadratic inequalities.

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127
g. Is it possible for a quadratic inequality not to have a real solution? Justify your
answer by giving a particular example.
Activity 6: Am I a Solution or Not?
Directions: Determine whether or not each of the following points is a solution of the
inequality 532 2
 xxy . Justify your answer.
1.  6,1A 6.  9,2F
2.  8,1B 7.  2,6 G
3.  10,5C 8.  4,1H
4.  6,3D 9.  0,0I
5.  4,3E 10.  7,6J
Were you able to able to find and describe the solution set of each quadratic
inequality? Were you able to show the graph of the solution set of each? In the
next activity, you will determine if a point is a solution of a given quadratic
inequality in two variables.
How did you find the activity? Was it easy for you to determine if a point is a
solution of the given inequality? Could you give other points that belong to the
solution set of the inequality? I’m sure you could. In the next activity, you will
determine the mathematical sentence that is described by a graph.

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128
Activity 7: What Represents Me?
Directions: Select from the list of mathematical sentences on the right side the
inequality that is described by each of the following graphs. Answer the
1.
2.
y
x
y
x
822
 xxy
822
 xxy
822
 xxy
572 2
 xxy
572 2
 xxy
822
 xxy
822
 xxy
572 2
 xxy
572 2
 xxy

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129
3.
4.
822
 xxy
822
 xxy
822
 xxy
822
 xxy
572 2
 xxy
572 2
 xxy
822
 xxy
572 2
 xxy
572 2
 xxy
y
x
y
x

DRAFT
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130
Questions:
a. How did you determine the quadratic inequality that is described by a given
graph?
b. In each graph, what does the shaded region represent?
c. How do the points in the shaded region of each graph facilitate in determining
the inequality that defines it?
d. How would you describe the graphs of quadratic inequalities in two variables
involving “less than”? “greater than”? “less than or equal to”? “greater than or
equal to”?
e. Suppose you are given a quadratic inequality in two variables. How will you
graph it?
Activity 8: Make It Real!
Directions: Read the situation below then answer the questions that follow.
1. How would you represent the width of the floor?
How about its length?
2. What mathematical sentence would represent the given situation?
3. What are the possible dimensions of the floor?
How about the possible areas of the floor?
The floor of a conference hall can be
covered completely with tiles. Its length is 36 ft.
longer than its width. The area of the floor is
less than 2,040 square feet.
Were you able to identify the inequality that is described by each graph? Was it
easy for you to perform this activity? I’m sure it was! If not, then try to find an
easier way of doing this activity. I know you can do it. In the next activity, you will
work on a situation involving quadratic inequalities. In this activity, you will further
learn how quadratic inequalities are illustrated in real life.

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131
4. Would it be realistic for the floor to have an area of 144 square feet? Explain
your answer.
Activity 9: How Well I Understood…
1. How do you describe quadratic inequalities?
2. Give at least three examples of quadratic inequalities.
3. How do you find the solution set of a quadratic inequality in one variable?
How about quadratic inequalities in two variables?
4. How would you describe the solution set of each of the following quadratic
inequalities?
a. 1492
 xxy c. 5112 2
 xxy
b. 1832
 xxy d. 8103 2
 xxy
are going to think deeper and test further your understanding of quadratic
inequalities. After doing the following activities, you should be able to answer the
following question: “How are quadratic inequalities used in solving real-life
problems and in making decisions?”
In this section, the discussion was about quadratic inequalities and their solution
sets and graphs.

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132
5. Do you agree that the solution sets of 202
 xxy and 202
 xxy is the
set of all points on a plane? Justify your answer by graphing the solution set of
each on a coordinate plane.
6. Luisa says that the solutions of 782 2
 xxy are also solutions of
342
 xxy . Do you agree with Luisa? Justify your answer.
7.
8. A company decided to increase the size of the box for the packaging of their
canned sardines. The length of the original packaging box was 40 cm longer than
its width, the height was 12 cm, and the volume was at most 4,800 cm3.
a. How would you represent the width of the original packaging box? How about
the length of the box?
b. What expression would represent the volume of the original packaging box?
How about the mathematical sentence that would represent its volume?
Define the variables used.
c. What could be the greatest possible dimensions of the box if each dimension
is in whole centimeters? Explain how you arrived at your answer.
d. Suppose the length of the new packaging box is still 40 cm longer than its
width and the height is 12 cm. What mathematical sentence would represent
the volume of the new packaging box? Define the variables used.
e. What could be the dimensions of the box? Give the possible dimensions of at
least three different boxes.
inequalities and their solution sets and graphs.
What new insights do you have about quadratic inequalities and their solution
sets and graphs? How would you connect this to real life? How would you use
A rectangular box is completely filled with
dice. Each die has a volume of 1 cm3. The
length of the box is 3 cm greater than its
width and its height is 5 cm. Suppose the
box holds at most 140 dice. What are the
possible dimensions of the box?

DRAFT
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133
What to TRANSFER:
Activity 10: Investigate Me!
Directions: Conduct a mathematical investigation for each of the following quadratic
inequalities. Prepare a written report of your findings following the format at the right.
1. 0cba 2
 xx , where 0ac4b2

2. 0cba 2
 xx , where 0ac4b2

3. 0cba 2
 xx , where 0ac4b2

4. 0cba 2
 xx , where 0ac4b2

Activity 11: How Much Would It Cost to Tile a Floor?
Directions: Perform the following activity.
1. Find the dimensions of the floors of at least two rooms in your school. Indicate
the measures obtained in the table below.
Rooms Length Width
be given a practical task which will demonstrate your understanding.
Mathematical Investigation Written
Report Format
Title/Topic For Investigation
Introduction
Exploration
Conjectures
Testing Of Conjectures
Justification For Conjectures
Conclusion

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134
2. Determine the measures and costs of different tiles that are available in the
nearest hardware store or advertised in any printed materials or in the internet.
Write these in the table below.
Tiles Length Width Cost
3. Formulate quadratic inequalities involving the dimensions of the floor of rooms,
and the measures and costs of tiles. Find, then graph the solution sets of these
inequalities. Use the rubric provided to rate your work.
Rubric for Real-Life Situations Involving Quadratic Inequalities and Their Solution
Sets and Graphs
4 3 2 1
Systematically
listed in the table
the dimensions of
rooms and the
measures and
costs of tiles,
properly formulated
and solved
quadratic
inequalities, and
accurately drew
the graphs of their
solution sets.
Systematically
listed in the table
the dimensions of
rooms and the
measures and
costs of tiles, and
properly formulated
and solved
quadratic
inequalities but
unable to draw the
graph accurately.
Systematically
listed in the table
the dimensions of
rooms and the
measures and
costs of tiles, and
properly formulated
quadratic
inequalities but
unable to solve
these.
Systematically
listed in the table
the dimensions of
rooms and the
measures and
costs of tiles.
In this section, your task was to formulate and solve quadratic inequalities based
on real-life situations. You were placed in a situation wherein you need to
determine the number and total cost of tiles needed to cover the floors of some
rooms.
How did you find the performance task? How did the task help you realize the
use of the topic in real life?

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135
This lesson was about quadratic inequalities and their solution sets and
graphs. The lesson provided you with opportunities to describe quadratic inequalities
and their solution sets using practical situations, mathematical expressions, and their
graphs. Moreover, you were given the opportunity to draw and describe the graphs of
quadratic inequalities and to demonstrate your understanding of the lesson by doing a
practical task. Your understanding of this lesson and other previously learned
mathematics concepts and principles will facilitate your learning of the next lesson,
Quadratic Functions.
GLOSSARY OF TERMS
1. Discriminant – This is the value of the expression ac4b2
 in the quadratic
formula.
2. Extraneous Root or Solution – This is a solution of an equation derived from an
original equation. However, it is not a solution of the original equation.
3. Irrational Roots – These are roots of equations which cannot be expressed as
quotient of integers.
4. Quadratic Equations in One Variable – These are mathematical sentences of
degree 2 that can be written in the form 0cba 2
 xx .
5. Quadratic Formula – This is an equation that can be used to find the roots or
solutions of the quadratic equation 0cba 2
 xx . The quadratic formula is
a
acbb
x
2
42

 .
6. Quadratic Inequalities – These are mathematical sentences that can be written in
any of the following forms: 0cba 2
 xx , 0cba 2
 xx , 0cba 2
 xx ,
and 0cba 2
 xx .
7. Rational Algebraic Equations – These are mathematical sentences that contain
rational algebraic expressions.
8. Rational Roots – These are roots of equations which can be expressed as
quotient of integers.

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136
9. Solutions or Roots of Quadratic Equations – These are the values of the
variable/s that make quadratic equations true.
10.Solutions or Roots of Quadratic Inequalities – These are the values of the
variable/s that make quadratic inequalities true.
DEPED INSTRUCTIONAL MATERIALS THAT CAN BE USED AS ADDITIONAL
RESOURCES FOR THE LESSON QUADRATIC EQUATIONS AND
INEQUALITIES:
1. EASE Modules Year II Modules 1, 2 and 3
2. BASIC EDUCATION ASSISTANCE FOR MINDANAO (BEAM) Mathematics 8
Module 4 pp. 1-55
REFERENCES AND WEBSITE LINKS USED IN THIS MODULE:
References:
Bellman, Allan E., et. Al. Algebra 2 Prentice Hall Mathematics, Pearson Prentice
Hall, New Jersey USA, 2004.
Brown, Richard G., Mary P. Dolciani, Robert H. Sorgenfrey and William L. Cole.
Algebra, Structure and Method, Book I, Houghton Mifflin Company, Boston MA,
1990.
Brown, Richard G., Mary P. Dolciani, Robert H. Sorgenfrey, and Robert B. Kane.
Algebra, Structure and Method Book 2. Houghton Mifflin Company, Boston,
1990.
Chapin, Illingworth, Landau, Masingila and McCracken. Prentice Hall Middle Grades
Math, Tools for Success, Prentice-Hall, Inc., Upper Saddle River, New Jersey,
1997.
Clements, Douglas H., Kenneth W. Jones, Lois Gordon Moseley and Linda
Schulman. Math in my World, McGraw-Hill Division, Farmington, New York,
1999.
Coxford, Arthur F. and Joseph N. Payne. HBJ Algebra I, Second Edition, Harcourt
Brace Jovanovich, Publishers, Orlando, Florida, 1990.

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Fair, Jan and Sadie C. Bragg. Prentice Hall Algebra I, Prentice-Hall, Inc., Englewood
Cliffs, New Jersey, 1991.
Gantert, Ann Xavier. Algebra 2 and Trigonometry. AMSCO School Publications, Inc.,
2009.
Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 1,
Applications, Equations, and Graphs. McDougal Littell, A Houghton Mifflin
Company, Illinois, 2004.
Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Algebra 2,
Applications, Equations, and Graphs. McDougal Littell, A Houghton Mifflin
Company, Illinois, 2008.
Smith, Charles, Dossey, Keedy and Bettinger. Addison-Wesley Algebra, Addison-
Wesley Publishing Company, 1992.
Wesner, Terry H. and Harry L. Nustad. Elementary Algebra with Applications. Wm.
C. Brown Publishers. IA, USA.
Wilson, Patricia S., et. al. Mathematics, Applications and Connections, Course I,
Glencoe Division of Macmillan/McGraw-Hill Publishing Company, Westerville,
Ohio, 1993.
WEBSITE Links as References and for Learning Activities:
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Beginning Algebra (v. 1.0) Section 6.6 Solving Equations by Factoring. (2012).
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algebra/s09-06-solving-equations-by-factoring.html
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algebra/s12-01-extracting-square-roots.html

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Beginning Algebra (v. 1.0) Section 9.2 Completing the Square. (2012). Retrieved
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Math-Help-Ace.com.Quadratic Equation Solver - Finding The Quadratic Equation
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adratic.html
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Seward, Kim. WTAMU and Kim Seward Virtual Math Lab College Algebra Tutorial
23A. Quadratic Inequalities. (2010). Retrieved November 7, 2013, from
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut23_
quadineq.htm
She Loves Math Quadratic Inequalities. (2013). Retrieved November 7, 2013, from
http://www.shelovesmath.com/algebra/intermediate-algebra/quadratic-inequalities/
Slideshare Inc. Quadratic Equation Word Problems. (2013). Retrieved November 7,
2013, from http://www.slideshare.net/jchartiersjsd/quadratic-equation-word-problems
Solving Quadratic Equations by Using the Square Root Property. Retrieved
November 7, 2013, from http://www.personal.kent.edu/~bosikiew/Algebra-
handouts/solving-sqroot.pdf
The Purple Math Forums Completing the Square Solving Quadratic Equations.
http://www.purplemath.com/modules/sqrquad.htm
The Purple Math Forums. Quadratic World Problems: Projectile Motion. (2012).
http://www.purplemath.com/modules/quadprob.htm
The Purple Math Forums Solving Quadratic Equations by Factoring. (2012).
http://www.purplemath.com/modules/solvquad.htm
The Purple Math Forums Solving Quadratic Equations by Taking Square Roots.
http://www.purplemath.com/modules/solvquad2.htm
The Purple Math Forums. The Quadratic Formula Explained. (2012). Retrieved
November 7, 2013, from http://www.purplemath.com/modules/quadform.htm
TutorVista.com. (2013). Algebra 1 Quadratic Equation. Retrieved November 7, 2013,
from http://math.tutorvista.com/algebra/quadratic-equation.html

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uwlax.edu. Retrieved November 7, 2013, from
http://www.uwlax.edu/faculty/hasenbank/archived/mth126fa08/notes/11.10%20-
%20Quadratic%20Applications.pdf
WEBSITE Link for Videos:
easymathk12. Retrieved November 7, 2013, from
http://www.youtube.com/watch?v=l7FI4T19uIA
Math Vids. Solving Rational Equation to Quadratic 1. (2013). Retrieved November 7,
2013, from http://mathvids.com/lesson/mathhelp/1437-solving-rational-equation-to-
quadratic-1
WEBSITE Links for Images:
Smith, Cameron. Kentucky girls basketball star scores all of her team’s first 40 points
in win. (2012). Retrieved November 7, 2013, from
http://sports.yahoo.com/blogs/highschool-prep-rally/kentucky-girls-basketball-star-
scores-her-team-first-181307906.html

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MODULE 2. QUADRATIC FUNCTIONS
I. Introduction
Have you ever asked yourself why PBA star players are good in free throws? How
do angry bird expert players hit their targets? Do you know the secret key in playing this
game? What is the maximum height reached by an object thrown vertically upward given a
particular condition?
One of the most interesting topics in mathematics is the quadratic function. It has
many applications and has played a fundamental role in solving many problems related to
human life. In this module, you will be able to learn important concepts in quadratic functions
which will enable you to answer the questions above. Moreover, you will also deal with the
most common applications of quadratic functions.
II. Lesson and Coverage
This module consists of five lessons namely:
Lesson 1- Introduction to Quadratic Functions
Lesson 2 - Graphs of Quadratic Functions
Lesson 3 - Finding the Equation of Quadratic Function
Lesson 4 - Applications of Quadratic Functions
In this module, you will learn to:
Lesson 1  model real life situations using quadratic functions
 differentiate quadratic functions from linear or other
functions.
 represent and identify the quadratic function given
- table of values
- graphs
- equation
 transform the quadratic function in general form y = ax2
+
bx + c into standard form (vertex form) y = a(x - h)2
+ k
and vice versa.
http://www.popularmechanics.com/cm/po
pularmechanics/images/y6/free-throw-
0312-mdn.jpg
http://web.mnstate.edu/lindaas/phys160/lab/S
ims/projectileMotion.gif -throw-0312-mdn.jpg
http://sites.davidson.edu/mathmovement/wp
content/uploads/2011/11/trajectories.png

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Lesson 2  draw the graph of the quadratic function
 given a quadratic function, determine the following:
domain, range, intercepts, axis of symmetry and the
opening of the parabola.
 investigate and analyze the effects of changes in the
variables a, h and k in the graph of quadratic functions
y = a(x – h)2
+ k and make generalizations.
 apply the concepts learned in solving real life problems.
Lesson 3  determine the zeros of quadratic functions
 derive the equation of the quadratic function given
- table of values
- graphs
- zeros
 apply the concepts learned in solving real life problems.
Lesson 4  solve problems involving quadratic function
Module Map
Applications of
Quadratic
Functions
Quadratic Functions
Introduction
to Quadratic
Functions
Forms of
Quadratic
Functions
Graph of the
Quadratic
Function
Finding the
Equations of
Quadratic
Functions
Properties of the
graph
Transformation of
the graph
Given:
Table of Values
Graph
Zeros

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III. Pre -Assessment
Part I. Find out how much you already know about this module. Write the letter that you think
is the best answer to each question on a sheet of paper. Answer all items. After taking and
checking this short test, take note of the items that you were not able to answer correctly and
look for the right answer as you go through this module.
1. Which of the following equations represents a quadratic function?
a. y = 3 + 2x2
b. 2y2
+ 3 = x
c. y = 3x - 22
d. y = 2x
– 3
2. The quadratic function f(x) = x2
+ 2x – 1 is expressed in standard form as
a. f(x) = ( x + 1 )2
+ 1 b. f(x) = ( x + 1 )2
– 2
c. f(x) = ( x + 1 )2
+ 2 d. f(x) = ( x + 1 )2
– 1
3. What is f(x) = -3( x + 2 )2
+ 2 when written in the form f(x) = ax2
+ bx + c?
a. f(x) = -3x2
+12 x -10 b. f(x) = 3x2
-12 x +10
c. f(x) = -3x2
+12 x +10 d. f(x) = -3x2
-12 x -10
4. The zeros of the quadratic function described by the graph below is
a. 1, 3
b. -1, 3
c. 1, -3
d. -1, -3
5. The graph of y = x2
– 3 is obtained by sliding the graph of y = x2
a. 3 units downward b. 3 units upward
c. 3 units to the right d. 3 units to the left
6. The quadratic function y = -2x2
+ 4x - 3 has
a. real and unequal zeros
b. real and equal zeros
c. no real zeros
d. equal and not real
7. What is an equation of a quadratic function whose zeros are twice the zeros of y = 2x2
–x
– 10?
a. f(x) = 2x2
-20 x +20 b. f(x) = x2
- x - 20
c. f(x) = 2x2
-2x -5 d. f(x) = 2x2
-2 x – 10
8. Which of the following shows the graph of f(x) = 2(x-1)2
– 3
a b. c. d.

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9. Richard predicted that the number of mango trees x planted in a farm could yield
y = -20x2
+ 2800x mangoes per year. How many trees should be planted to produce the
maximum number of mangoes per year?
a. 60 b.70 c.80 d.90
10. The path of an object when it is thrown can be modeled by S(t) = -16t2
+ 8t +4 where S in
feet is the height of the object t seconds after it is released. What is the maximum height
reached by the object?
a. 3 ft b. 4 ft c. 5 ft d. 6 ft
11. CJ wrote a function of the path of the stone kicked by Lanlan from the ground. If the
equation of the function he wrote is S(t) = 16t2
+ 8t +1, where S is the height of stone in
terms of t the number of seconds after Lanlan kicks the stone. Which of the statement is
true?
a. CJ’s equation is not correct.
b. CJ’s equation described the maximum point reached by the stone.
c. The equation is possible to the path of the stone.
d. The equation corresponds to the path of the stone.
12. An object is fired straight up with a velocity of 64 ft/s. Its altitude (height) h after t
seconds is given by h(t) = −16t2
+ 64t. When does the projectile hit the ground?
a. 3 seconds b. 4 seconds c. 5 seconds d. 6 seconds
13. What are the dimensions of the largest rectangular field that can be enclosed with 100m
of wire?
a. 24 m x 26 m b. 25m x 25 m
c. 50m x 50m d. 50m x 25 m
14. The batter hits the softball and it follows a path in which the height h is given by h(t) =-2t2
+ 8t +3, where t is the time in seconds s elapsed since the ball was pitched. What is the
maximum height reached by the softball?
a. 11 m b. 12m c. 13m d. 14m
Part II Performance Task
Directions: Apply quadratic functions to solve the problem below. Show your
solution.
Task 1 Being the first grandson, your grandparents decided to give you a
rectangular field for your coming wedding. If you are given 200m wires of fencing,
what dimensions would you choose to get the maximum area?
a. List all the possible dimensions of the rectangular field.
b. Make a table of values for the possible dimensions.
c. Compute the area for each possible dimension.
d. What is the maximum area you obtained?
e. What are the dimensions of the maximum area you obtained?

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Task 2 You are selling banana bread that costs P 5 each. Each week, you have 50
customers. When you decrease the price by P1, you expect 30 customers to be
added. What is the price of the banana bread that yields a maximum profit?
a. Analyze the problem.
b. What is the weekly sale if the cost of the banana bread is P5?
c. If the revenue (R) = number of bread x bread price. Write the equation of
the quadratic function given the situation above?
d. What is the price that yields the maximum revenue?
e. Find the maximum revenue.
IV LEARNING GOALS AND TARGETS
After going through this module, you should be able to demonstrate understanding of
the key concepts of quadratic functions and be able to apply these to solve real life
problems. You will be able to formulate real life problems involving quadratic functions, and
solve them through a variety of techniques with accuracy.
V INSTRUCTIONAL ACTIVITIES
Lesson 1. Introduction to Quadratic Function
What to KNOW
Let us start this lesson by recalling ways of representing a linear
function. The knowledge and skills in doing this activity will help you a lot in
understanding the quadratic function. In going over this lesson, you will be
able to identify quadratic function and represent it in different ways.

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Activity 1. Describe me in many ways!
Directions: Perform this activity.
a. Observe the pattern and draw the 4th
and 5th
figures.
? ?
1 2 3 4 5
b. Use the table to illustrate the relation of the figure number to the number of blocks.
c. Write the pattern observed from the table. _______________
d. List the following:
Set of ordered pairs_______________
Domain _______________ Range _______________
e. What equation describes the pattern? _______________
f. Graph the relation using the Cartesian Plane.
g. What are the independent and dependent variables? _______________
h. What methods are used to describe the relation? _______________
Figure Number (x) 1 2 3 4
5
Number of blocks (y) 1 4 7

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Activity 2 Parking Lot Problem
Directions: Solve the problem by following the procedure below.
Mr Santos wants to enclose the rectangular parking lot beside his house by putting a
wire fence on the three sides as shown in the figure. If the total length of the wire is 80 m,
find the dimension of the parking lot that will enclose a maximum area.
Directions. Follow the procedure below:
a. In the figure above, if we let w be the width and l be the length, what is the
expression for the sum of the measures of the three sides of the parking lot?
b. What is the length of the rectangle in terms of the width?
c. Express the area (A) of the parking lot in terms of the width.
d. Fill up the table by having some possible values of w and the corresponding areas.
Width (w)
Area (A)
e. What have you observed about the area (A) in relation to the width (w)?
f. What is the dependent variable? independent variable?
g. Compare the equation of a linear function with the equation you obtained.
Parking lot
Housew
l
To the graphic
artist, kindly put
fences on the three
sides of the
parking lot

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149
h. From the table of values, plot the points and connect them using a smooth curve.
i. What do you observe about the graph?
j. Does the graph represent a linear function?
Activity 3 Identify me!
Directions: State whether each of the following equations represents a quadratic function or
not. Justify your answer.
Equations Yes
or No
Justification
1. y = x2
+ 2
2. y = 2x – 10
3. y = 9 – 2x2
4. y = 2x
+ 2
5. y = 3x2
+ x3
+ 2
6. y = 2x
+3x + 2
7. y = 2x2
8. y = (x- 2) (x + 4)
9. 0 = (x- 3)(x+3)+ x2
-y
10. 3x3
+ y – 2x = 0
How did you find the preceding activity? I hope that you are now ready
to learn about quadratic functions. These are functions that can be described
by equations of the form y = ax2
+ bx + c, where, a, b and c are real numbers
and a  0. The highest power of the independent variable x is 2. Thus, the
equation of a quadratic function is of degree 2.

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Activity 4 Compare Me!
Directions: Follow the instructions below.
Consider the given functions f(x) = 2x + 1 and g(x) = x2
+ 2x -1.
1. What kind of function is f(x)? g(x)?
2. Complete the following table of values using the indicated function
f(x) = 2x + 1 g(x) = x2
+ 2x -1
3. What are the differences between two adjacent x-values in each table?
4. Find the differences between each adjacent y- values in each table, and write them
on the blanks provided.
f(x) = 2x + 1 g(x) = x2
+ 2x -1
5. What do you observe?
6. How can you recognize a quadratic function when a table of values is given?
7. Using the table of values, graph the two functions and compare the results.
y = 2x + 2 y = x2
+ 2x -1
x -3 -2 -1 0 1 2 3
y
xx -3 -2 -1 0 1 2 3
y
x -3 -2 -1 0 1 2 3
y
x -3 -2 -1 0 1 2 3
y

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8. Compare the graph of linear function and quadratic function.
In a quadratic function, equal differences in the independent variable x produce equal
second differences in the dependent variable y.
Illustrative example
Let us consider y = x2
– 4
Differences in x
First Differences in y
Second Differences in y
Study the illustrative examples presented below.
Example 1
Express y = 3x2
– 4x + 1 in the form y = a(x - h)2
+ k form and give the values of h
and k.
x -3 -2 -1 0 1 2 3
y 5 0 -3 -4 -3 0 5
Did you enjoy the activity? You have seen that in a linear
function, equal differences in x produce equal differences in y. However,
in a quadratic function, equal differences in x do not lead to equal first
differences in y; instead the second differences in y are equal. Notice
also that the graph of a linear function is a straight line while the graph of
a quadratic function is a smooth curve. This smooth curve is parabola.
-5 -3 -1 1 3 5
2 2 2 2 2
1 1 1 1 1 1
You have seen in the example above that in the quadratic
function y = ax2
+ bx + c, equal differences in x produce equal
second differences in y.
The previous activities familiarized you with the general form
y = ax2
+ bx + c of a quadratic function. In your next activity, the
standard form or vertex form y = a(x - h)2
+ k will be introduced. The
standard form will be more convenient to use when working on problems
involving the vertex of the graph of a quadratic function.

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152
Solution:
y = 3x2
– 4x + 1
y = (3x2
– 4x) + 1 Group together the terms containing x
y = 3(x2
–
3
4
x) + 1 Factor out a. Here a = 3
y = 3















2
2
3
2
3
4
xx + 1 - 3
2
3
2






Compete the expression in parenthesis to
make it a perfect square trinomial by adding the
constant
3
9
4
3
3
2
3
2
3
4
2
2



















=
3
4
and subtracting the
same value from the constant term.
y = 3 






9
4
3
42
xx + 1 - 





3
4
Simplify and express the perfect square
trinomial as square of binomial
y = 3
2
3
2






x -
3
1
Hence, y = 3x2
– 4x + 1 can be expressed as y = 3
2
3
2






x -
3
1
.
In this case, h =
3
2
and k = -
3
1
.
Example 2
Rewrite f (x) = ax2
+bx + c in the form f(x) = a(x - h)2
+ k.
y =   cbxax 2
Group together the terms containing x
y = a cx
a
b
x 





2
Factor out a. Here a = 1.
y = a
a
b
c
a
b
x
a
b
x
44
2
2
2
2






 Compete the expression in the parenthesis to
make it a perfect square trinomial by adding a
constant a
2
2






a
b
=
a
b
4
2
and subtracting the
same value from the constant term
y = a
a
bac
a
b
x
4
4
2
22







 Simplify and express the perfect square
trinomial as square of binomial

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153
Hence, the vertex form is y = a
a
bac
a
b
x
4
4
2
22







 . Thus, h =
a
b
2

and k =
a
bac
4
4 2

.
Example 3
Rewrite f (x) = x2
– 4x – 10 in the form f(x) = a(x - h)2
+ k.
Solution 1
By completing the square:
y =   1042
 xx Group together the terms containing x
y =   1042
 xx Factor out a. Here a = 1.
y =   410442
 xx Compete the expression in parenthesis to
make it a perfect square trinomial by adding a
constant
2
2
4





 
= 4 and subtracting the same
value from the constant term.
y =   142
2
x Simplify and express the perfect square
trinomial as square of binomial.
Solution 2
By applying the formula h =
a
b
2

and k =
a
bac
4
4 2

:
In the equation y = x2
– 4x – 10, a = 1, b = -4 and c = -10. Thus,
h =
a
b
2

k =
a
bac
4
4 2

h =
)1(2
)4(
k =
)1(4
)4()10)(1(4 2

h =
2
4
k =
4
1640 
h = 2 k = -14
By substituting the solved value of h and k in y = a(x - h)2
+ k, we obtain
y = (x - 2)2
- 14 .

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154
Activity 5 Step by step!
Directions: Work in pairs. Transform the given quadratic functions into the form y = a(x - h)2
+ k by following the steps below.
1. y = x2
– 4x – 10
2. y = 3x2
– 4x + 1
Steps Task
1. Group the terms containing x
2. Factor out a
3. Complete the expression in
parenthesis to make it a perfect square
trinomial.
4. Express the perfect square trinomial as
square of a binomial
1. 5. Give the value of h
2. 6. Give the value of k
To transform a quadratic function from standard form y = a(x - h)2
+ k into
general form, consider the examples below.
Example 4
Rewrite the equation y = 3( x- 2)2
+ 4 in the general form y = ax2
+ bx + c.
Solution:
y = 3( x- 2)2
+ 4 Expand (x – 2)2
.
y = 3( x2
– 4x + 4) +4 Multiply the perfect square trinomial
by 3.
y = 3x2
– 12x + 12 + 4 Simplify and add 4.
y = 3x2
– 12x + 16
Example 5
Express f(x) = - 2( 3x - 1 )2
+ 5x in the general form f(x) = ax2
+ bx + c.
Did you transform the quadratic function in the form y = a(x - h)2
+ k?

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155
Solution:
f(x) = - 2( 3x - 1 )2
+ 5x
f(x) = - 2( 9x2
- 6x + 1 ) + 5x
f(x) = - 18x2
+ 12x - 2 + 5x
f(x) = - 18x2
+ 17x - 2
Activity 6 Reversing the process
A. Directions: Rewrite y = 2( x- 1)2
+ 3 in the form y = ax2
+ bx + c by following the
given steps.
Steps Task
1. Expand (x – 1)2
2. Multiply the perfect square trinomial
by 2
3. Simplify
4. Add 3
3. 5. Result
B. Apply the above steps in transforming the following quadratic functions into the
general form.
1. y = 2(x - 4 )2
+ 5
2. y = 3 






2
1
x x + 1
What to PROCESS
Your goal in this section is to master the skills in identifying the
quadratic function and transforming it in different forms. Towards the end of
this module, you will be encouraged to apply these skills in solving real- life
problems.
Did you transform the quadratic function into the form y = ax2
+bx + c?

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156
Activity 7 Where do you belong?
A. Put the letter of the given equation in the diagram below where you think it
belongs.
a. y = x2
– 1 f. y = x3
+ 1
b. y = x g. 22
+ x = y
c. 2x2
– 2x + 1 = y h. y = 3x
+ 2x
d. 3x-1 + y = 0 i. 3x +x2
= y
e. y = (2x+ 3) ( x- 1) j. 2x(x – 3) - y = 0
How did you classify each of the given functions?
What similarities do you see in quadratic functions? in linear functions?
How do a quadratic function and a linear function differ?
What makes a function quadratic?
Quadratic
function
Linear
function
Not linear nor
Quadratic

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157
Activity 8 Quadratic or Not
Directions: Study the patterns below. Indicate whether the pattern described by the figures
is quadratic or not.
a.
Determine the relationship between the number of blocks in the bottom row and the total
number of blocks.
What relationship exists between the two numbers?
b.
number of blocks.
c.
number of blocks in the figure.

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158
Activity 9 It’s your turn
Direction: Match the given quadratic function y = ax2
+ bx + c to its equivalent standard form
y = a(x - h)2
+ k.
What mathematical concepts did you use in doing the transformation?
Explain how the quadratic function in the form y = ax2
+ bx +c can be transformed into the
form y = a(x - h)2
+ k.
y = x2
– x +
4
13
y =
2
1
x2
–3x + 3
y = -2x2
+12x -17
y = x2
– 4x + 1
y = 2x2
– 4x +4
y = (x - 2)2
- 3
y = 2(x-1)2
+ 2
y = -2(x-3)2
+1
y =
2
1
(x- 3)2
-
2
3
y = (x-
2
1
)2
+ 3

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Activity 10 The hidden message
Directions: Write the indicated letter of the quadratic function in the form y = a(x-h)2
+ k into
the box that corresponds to its equivalent general form y = ax2
+ bx + c.
I y = (x - 1)2
-4 T y = (x – 1)2
-16
S y = 2
8
49
4
5
2






x F y = (x-3)2
+ 5
E y = 2
3
2
2






x M y =
2
3
2
1
2






x
A y = 3(x + 2)2
-
2
1
U y = -2(x -3)2
+ 1
N y = (x - 0)2 – 36 H y = 2(x + 1)2
- 2
DIALOG BOX:
y = x2
–x +
4
7
y = 3x2
+ 12x +
2
23
y = x2- 2x – 15
y = 2x2
+ 4x
y = x2 - 2x -3
y = 2x2
+5x -3

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y = x2
– 6x + 14
y = -2x2
+ 12x -17
y = x2
- 36
How is the square of a binomial obtained without using the long method of
multiplication?
Explain how the quadratic function in the form y = a(x - h)2
+ k can be transformed
into the form y = ax2
+ bx +c.
Activity 11 Hit or Miss!
Direction: Work in Pairs. Solve this problem and show your solution.
Problem. An antenna is 5 m high and 150 meters from the firing place. Suppose the path of
the bullet shot from the firing place is determined by the equation y = - 2
1500
1
x + x
15
2
,
where x is the distance (in meters) of the bullet from the firing place and y is its height. Will
the bullet go over the antenna? If yes/no, show your justification.
What to REFLECT or UNDERSTAND
Your goal in this section is to have a better understanding of the
mathematical concepts about quadratic functions. The activities provided
for you in this section aim to apply the different concepts that you have
learned from the previous activities.

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Activity 12 Inside Outside Circle (Kagan, 1994)
Directions:
Guide Questions/Topics for the activity.
1. What is a quadratic function?
2. How do you differentiate the equation of a quadratic function from that of a linear
function?
3. Describe the graph of a linear function and the graph of a quadratic function.
4. Given a table of values, how can you determine if the table represents a quadratic
function?
Activity 13. Combination Notes
A. In the oval callout, describe the ways of recognizing a quadratic function.
1. Form a group of 20 members
and arrange yourselves by following
the formation below.
2. Listen to your teacher regarding
the procedures of the activity.
Quadratic
Functions
Graph
EquationTable of Values

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B. In the oval callout, make an illustrative example of the indicated mathematical
concept.
Activity 14 Find my pattern!
Directions: Group yourselves into 5. Perform the activity below.
Consider the set of figures below. Study the relationship between the term number
and the number of unit triangles formed. What is the pattern? Describe the patterns through
a table of values, graph and equation. How many triangles are there in the 25th
term?
Term number 1 2 3
What to TRANSFER
Based on what you have learned in the preceding activity, you are now
ready to apply the concepts that you have learned in other contexts.
The goal of this section is for you to apply what you have learned in a
real-life situation. You will be given a task which will demonstrate your
understanding of the lesson.
Concepts Illustrative Example
Transforming a quadratic
function in the form y =
ax2
+ bx + c into the form
y = a(x - h)2
+ k.
Transforming a quadratic
function in the form y =
a(x - h)2
+ k into the form
y = ax2
+ bx + c.

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Activity 15 Investigate!
Problem. You are given 50 m of fencing materials. Your task is to make a
rectangular garden whose area is maximum. Find the dimensions of such a rectangle.
Explain your solution.
Activity 16 Explore more!
Give at least three parabolic designs that you see in your community. Then create
your own design.
SUMMARY/SYNTHESIS/GENERALIZATION
This lesson introduced quadratic function. The lesson provided you with opportunities
to describe a quadratic function in terms of equation, graph, and table of values. You were
given a chance to compare and see the difference between quadratic functions and linear
functions or other functions.

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Lesson 2. Graphs of Quadratic Functions
What to KNOW
Activity 1 Describe my paths!
Directions: Follow the procedure in doing the activity.
a. Given the quadratic functions y = x2 - 2x – 3 and y = -x2 + 4x -1, transform
them into the form y = a( x - h)2 + k.
y = x2 - 2x – 3 y = -x2 + 4x -1.
b. Complete the table of values for x and y.
y = x2 - 2x – 3 y = -x2 + 4x -1
c. Sketch the graph on the cartesian plane.
x -3 -
2
-1 0 1 2 3 4 5
y
x -4 -
3
-2 -1 0 1 2 3
y
Let’s start this lesson by generating table of values of quadratic
functions and plotting the points on the coordinate plane. You will
investigate the properties of the graph through guided questions. As you
go through this lesson, keep on thinking about this question: How can the
graph of quadratic function be used to solve real life problems?

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d. What have you observed about the opening of the curves? Do you have any
idea where you can relate the opening of the curves?
e. Which of the 2 quadratic functions has a minimum point? maximum point?
Indicate below.
f. Observe each graph. Can you draw a line that divides the graph in such a
way that one part is a reflection of the other part? If there is any, determine
the equation of the line?
g. Take a closer look at the minimum point or the maximum point and try to
relate it with the values of h and k in the equation y = a(x – h)2 + k of the
function. Write your observations.
h. Can you identify the domain and range of the functions?
y = x2 - 2x – 3 Domain: __________ Range: ___________
y = -x2 + 4x -1 Domain: __________ Range: ___________
The graph of a quadratic function y = ax2 + bx + c is called parabola. You
have noticed that the parabola opens
upward or downward. It has a turning
point called vertex which is either the
lowest point or the highest point of the
graph. If the value of a > 0, the
parabola opens upward and has a
minimum point. If a < 0, the parabola
opens downward and has a maximum
point. There is a line called the axis of
symmetry which divides the graph into
two parts such that one-half of the graph
is a reflection of the other half.
If the quadratic function is expressed in
the form y = a(x - h)2 + k, the vertex is the point (h, k). The line x = h is the axis of
symmetry and k is the minimum or maximum value of the function.
The domain of a quadratic function is the set of all real numbers. The range
depends on whether the parabola opens upward or downward. If it opens upward,
the range is the set  kyy : ; if it opens downward, then the range is the set
 kyy : .
Quadratic Function Vertex (Turning Point) Maximum or Minimum
Point
y = x2 - 2x – 3
y = -x2 + 4x -1
Did you enjoy the activity? To better understand the properties of the
graph of quadratic function, study some key concepts below.
Vertex
Maximum Point
Axis of symmetry
Vertex
Minimum PointAxis of symmetry

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Activity 2 Draw me!
Directions: Draw the graph of the quadratic function y = x2 – 4x + 1 by following the
steps below.
1. Find the vertex and the line of symmetry by expressing the function in the
form y = a( x - h)2 + k or by using the formula h =
a
b
2

; k =
a
bac
4
4 2

if the
given quadratic function is in general form.
2. On one side of the line of symmetry, choose at least one value of x and
compute the value of y.
Coordinates of points: _____________________________________________
3. Similarly, choose at least one value of x on the other side and compute the
value of y.
Coordinates of points: _____________________________________________
4. Plot the points and connect them by smooth curve.
Activity 3 Play and Learn!
Directions: Work in a group of 5 members. Solve the puzzle and do the activity.
Problem: Think of a number less than 20. Subtract this
number from 20 and multiply the difference by twice the
original number. What is the number that will give the largest
product?
The first group who gives the largest product wins the game.”
a. Record your answer on the table below:
b. Draw the graph.
c. Find the vertex and compare it to your answer in the puzzle.
d. If n is the number you are thinking, then how can you express the other number,
which is the difference of 20 and the number you are thingking?
e. What is the product (P) of the two numbers? Formulate the equation.
f. What kind of function is represented by the equation?
g. Express it in standard form.
h. What is the largest product?
i. What is the number that that will give the largest product?
j. Study the graph and try to relate the answer you obtained in the puzzle to the
vertex of the graph. Write your observation.
Number (n)
Product (P)

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Activity 4 To the left to the right! Put me up put me down!
Directions: Form groups of 5 members each and perform this activity.
A. Draw the graphs of the following quadratic functions on the same coordinate
plane.
1. y = x2
2. y = 2x2
3. y = 3x2
4. y =
2
1
x2
5. y =
3
1
x2
6. y = - x2
7. y = - 2x2
a. Analyze the graphs.
b. What do you notice the shape of the graph of the quadratic function
y = ax2?
c. What happens to the graph as the value of a becomes larger?
d. What happens when 0 < a < 1?
e. What happens when a < 0 ? a > 0 ?
f. Summarize your observations.
B. Draw the graphs of the following functions.
1. y = x2
2. y = ( x -2 )2
3. y = ( x + 2 )2
4. y = ( x + 1 )2
5. y = ( x - 1 )2

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b. What do you notice about the graphs of quadratic functions whose
equations are of the form y = ( x – h )2?
c. How would you compare the graph of y = ( x – h )2 and that of y = x2?
d. Discuss your ideas and observations.
C. Draw the graphs of the following quadratic functions
1. y = x2
2. y = x2 + 2
3. y = x2 - 2
4. y = x2 - 3
5. y = x2 + 3
b. What do you notice about the graphs of quadratic functions whose
equations are of the form y = x2 + k?
c. How would you compare the graph of y = x2 + k and that of y = x2 when the
vertex is above the origin? below the origin?
d. What conclusion can you give based on your observations?
D. Draw the graphs of the following quadratic functions:
1. y = ( x – 2)2 + 4
2. y = ( x + 3)2 – 4
3. y = ( x – 1)2 – 3
4. y = ( x + 4)2 + 5
5. y = ( x + 2)2 - 2

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b. What is the effect of the variables h and k on the graph of y = ( x – h ) 2 + k
as compared to the graph of y = x 2?
c. Make your generalization on the graph of y = (x - h)2 + k.
In the graph of y = ax2 + bx + c, the larger the a is, the
narrower is the graph.
For a >0, the parabola opens upward.
To graph y = a ( x – h )2, slide the graph of y = ax2 horizontally h
units. If h > 0, slide it to the right, if h < 0, slide it to the left. The
graph has vertex ( h, 0 ) and its axis is the line x = h.
To graph y = ax2 + k, slide the graph of y = ax2 vertically k
units. If k > 0 slide it upward; if k < 0, slide it downward. The
graph has vertex ( 0, k ) and its axis of symmetry is the line x = 0
(y – axis).
To graph y = a ( x – h )2 + k, slide the graph of y = ax2
horizontally h units and vertically k units. The graph has a vertex
(h, k) and its axis of symmetry is the line x = h.
If a < 0, the parabola opens downward. The same
procedure can be applied in transforming the graph of a
quadratic function.
Did you enjoy the activity? To better understand the
transformation of the graph of quadratic function. Read some key
concepts below.

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Vertex of the graph of a quadratic function:
In standard form f(x) = a(x – h)2 + k, the vertex (h, k) can be directly obtained
from the values of h and k.
In general form f(x) = ax2 + bx + c, the vertex (h, k) can be obtained using the
formulas h =
a
b
2

and k =
a
bac
4
4 2

.
What to PROCESS
Activity 5 Draw and describe me!
Directions: Sketch the graph of each quadratic function and identify the vertex,
domain, range, and the opening of the graph, State whether the vertex is a minimum
or a maximum point, and write the equation of its axis of symmetry.
1. f(x) = x2
2. f(x) = 2x2 + 4x –3
Your goal in this section is to apply the mathematical concepts that
you have learned in graphing quadratic functions. Use these mathematical
concepts to perform the provided activities in this section.
vertex _____________
opening of the graph _____________
Vertex is a _____________ point
Equation of the axis of symmetry ______
Domain: _____ Range: _____
vertex _____________
Equation of the axis of symmetry ____

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3. f(x) =
2
1
x2 + 2
4. f(x) = -x2 – 2x –3
5. f(x) = (x + 2)2 + 3
vertex _____________
vertex _____________
vertex _____________

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6. f(x) = 2(x – 2)2
7. f(x) = -2x2 – 2
How did you find the activity? Explain the procedure on how to draw the graph
of a quadratic function.
Activity 6 Hit the volleyball
Problem. Carl Allan hit the volleyball at 3 ft above the ground
with an initial velocity of 32 ft/sec. The path of the ball is given by
the function S(t) = -16t2 + 32t + 3, where S is the height of the ball
at t seconds. What is the maximum height reached by the ball?
a. What kind of function is used to model the path of the volleyball?
b. Draw the path of the volleyball and observe the curve.
vertex _____________
vertex _____________

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173
c. What is the maximum height reached by the ball?
d. What is represented by the maximum point of the graph?
Activity 7 Match or mismatch!
Directions: Decide whether the given graph is a match or a mismatch with the
indicated equation of quadratic function. Write match if the graph corresponds with
the correct equation. Otherwise, indicate the correct equation of the quadratic
function.
1.
2.
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
y = (x +4)2
x
y
y = 2x2
-3
y
x

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3.
4.
5.
Share the technique you used to determine whether the graph and the equation of
the quadratic function are match or mismatch.
What characteristics of a quadratic function did you apply in doing the activity?
__________________________________________
__________________________________________
__________________________________________
y = 2x2
-12x +18
y
x
__________________________________________
__________________________________________
__________________________________________
y = x2
-12x +36
y
x
__________________________________________
__________________________________________
__________________________________________
y = -2x2
+ 4

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175
Activity 8 Translate Me!
Directions: The graph of f(x) = 2x2 is shown below. Based on this graph, sketch the
graphs of the following quadratic functions in the same coordinate system.
f(x) = 2x2 + 3 f. f(x) = 2(x + 4)2
a. f(x) = 2(x – 3)2
– 1 g. f(x) = -2x2
- 1
b. f(x) = 2(x + 2)2
h. f(x) = 2(x + 3)2
+ 5
c. f(x) = -2x2
+ 3 i. f(x) = 2(x – 3)2
- 2
d. f(x) = 2(x + 1)2
– 2 j. f(x) = - 2(x – 1)2
How did you find the activity?
Describe the movement of the graph for each quadratic function.
Your goal in this section is to have a deeper understanding of the graph of
quadratic functions. The activities provided for you in this section will be of great
help to enable you to apply the concepts in different contexts.
y
x

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Activity 9 Let’s analyze!
Directions: Analyze the problem and answer the given questions.
Problem 1: A ball on the playing ground was kicked by Carl Jasper. The parabolic
path of the ball is traced by the graph below. Distance is given in meters.
Questions:
a. How would you describe the graph?
b. What is the initial height of the ball?
c. What is the maximum height reached by the ball?
d. Determine the horizontal distance that corresponds to the maximum
distance.
e. Approximate the height of the ball after it has travelled 2 meters
horizontally.
f. How far does the ball travel horizontally before it hits the ground?
Problem 2: The path when a stone is thrown can be modelled by
y = -16x2 + 10x + 4, where y (in feet) is the height of the stone x seconds after it is
released.
a. Graph the function.
b. Determine the maximum height reached by the stone.
c. How long will it take the stone to reach its maximum height?
x
y
Height(in
meters)
T
T
Horizontal distance (in meters)

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Activity 10 Clock Partner Activity
Directions: Use your clock partner to discuss the following questions.
Activity 11 Combination Notes
Directions: Tell something about what you have learned.
Time Topics/Questions to be discussed/answered
2:00 How can you describe the graph of a quadratic function?
5:00 Tell something about the axis of symmetry of a parabola.
12:00 When do we say that the graph has a minimum/maximum value or point?
3:00 What does the vertex imply?
5:00 How can you determine the opening of the parabola?
6:00 How would you compare the graph of y =a ( x – h )2 and that of y = ax2?
11:00 How would you compare the graph of y = x2 + k with that of y = x2 when
the vertex is above the origin? below the origin?
1:00 How do the values of h and k in y = a(x-h)2 + k affect the graph of y =
ax2?
Write your name on your clock. Make an appointment with 12 of your classmates ,
one for each hour on the clock. Be sure you both record the appointment on your
clock. Only make an appointment if there is an open slot at that hour on both your
clock.
What are the
properties of the graph
of a quadratic
function? Discuss
briefly.
Enumerate the steps in
graphing a quadratic
function.
How can you
determine the vertex
of a quadratic
function?

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Activity 12 Which of which?
Directions: Answer each of the following questions.
1. Which has the larger area?
a. A rectangle whose dimensions are 25 m by 20 m
b. The largest possible area of a rectangle to be enclosed if the perimeter
is 50 m
2. Which has the lower vertex?
a. y = x2
+ 2x + 3
b. y = x2
– 4x+ 7
3. Which has the higher vertex?
a. y = -x2
– 6x + 15
b. y = - x2
+ 6
Reflections

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Activity 13: ABC in Math!
Directions: Play and Learn with this activity.
In activity 3, you have learned the effects of variables a, h, and k in the graph
of y = a(x – h)2 + k as compared to the graph of y = ax2. Now, try to investigate the
effect of variables a, b and c in the graph of quadratic function y = ax2 + bx + c.
What to TRANSFER
Activity 14 Quadratic design
GOAL: Your task is to design a curtain in a small restaurant that involves a
quadratic curve.
ROLE: Interior Designer
AUDIENCE: Restaurant Owner
SITUATION: Mr. Andal, the owner of a restaurant wants to
impress some of the visitors, as target clients, in the coming
wedding of his friend. As a venue of the reception, Mr Andal
wants a new ambience in his restaurant. Mr Andal requested
Did you enjoy the activities? I hope that you learned a lot in this section and
you are now ready to apply the mathematical concepts you gained from all the
activities and discussions.
In this section, you will be given a task wherein you will apply what you have
learned in the previous sections. Your performance and output will show
evidence of learning.
Results/
Observations
s

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180
you, as interior designer, to help him to change the interior of the restaurant
particularly the design of the curtains. Mr Andal wants you to use parabolic curves in
your design. Map out the appearance of the proposed design for the curtains in his
20 by 7 meters restaurant and estimate the approximate budget requirements for the
cost of materials based on the height of the design curve.
PRODUCT: Proposed plan for the curtain including the proposed budget based on
the original garden.
STANDARDS FOR ASSESSMENT:
You will be graded based on the rubric designed suitable for your task and
performance.
Activity 15 Webquest Activity. Math is all around.
Directions: Make a simple presentation of world famous parabolic arches.
Task:
1. Begin the activity by forming a group of 5 members.
Choose someone you can depend on to work
diligently and to do his fair share of work.
2. In your free time, start surfing the net for world
famous parabolic arches. As you search, keep a
record of where you go, and what you find on the
site.
3. Complete the project by organizing the data you
collected, including the name of the architect and the
purpose of creating the design.
4. Once you have completed the data, present it to the
class in a creative manner. You can use any of the
following but not limited to them.
 Multimedia presentation
 Webpages
 Posters
5. You will be assessed based on the rubric for this activity.
SUMMARY/SYNTHESIS/GENERALIZATION
This lesson was about graphs of quadratic functions. The lesson was able to equip
you with ample knowledge on the properties of the graph of quadratic functions. You were
made to experience graphing quadratic functions and their transformation. You were given
opportunities to solve real life problems using graphs of quadratic functions and to create a
design out of it.

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Lesson 3. Finding the Equation of a Quadratic Function
What to KNOW
Activity 1. Give me my roots!
Directions: Given a quadratic equation x2 –x - 6 =0, find the roots in three
methods.
A value of x that satisfies the quadratic equation ax2 + bx +c = 0 is called a
root of the equation.
Let’s begin this lesson by recalling the methods of finding the roots
of quadratic equations. Then relate them with the zeros of the quadratic
functions. In this lesson, you will be able to formulate patterns and
relationship regarding quadratic functions. Furthermore, you will be able to
solve real-life problems involving equations of quadratic functions.
Factoring
Quadratic
Formula
Did you find the roots in 3 different ways? Your skills in
finding the roots will also be the methods you will be using in finding
the zeros of quadratic functions. To better understand the zeros of
quadratic functions and the procedure in finding them, study the
mathematical concepts below.

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182
x
y
Activity 2 What are my zeros?
Directions: Perform this activity and answer the guided questions.
Examine the graph of the quadratic function y = x2 - 2x – 3
a. How would you describe the graph?
b. Give the vertex of the parabola and its axis of symmetry.
c. At what values of x does the graph intersect the x- axis?
d. What do you call these x- coordinates where the curve crosses the x- axis?
e. What is the value of y at these values of x?
The graph of a quadratic function is a parabola.
A parabola can cross the x-axis once, twice, or never.
The x-coordinates of hese points of intersection are
called x-intercepts. Let us consider the graph of the
quadratic function y = x2 - x – 6. It shows that the
curve crosses the x-axis at 3 and -2. These are the x-
intercepts of the graph of the function. Similarly, 3 and
-2 are the zeros of the function since these are the
values of x when y equals 0. These zeros of the function can be determined by
setting y to 0 and solving the resulting equation through different algebraic methods.
How did you find the activity? To better understand the zeros
of a function, study some key concepts below.
x
y

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183
Example 1.
Find the zeros of the quadratic function y = x2
– 3x + 2 by factoring method.
Solution:
Set y = 0. Thus,
0 = x2
– 3x + 2
0 = ( x – 2) ( x – 1)
x – 2 = 0 or x – 1 = 0
Then x = 2 and x = 1
The zeros of y = x2
– 3x + 2 are 2 and 1.
Example 2.
Find the zeros of the quadratic function y = x2 + 4x – 2 using the completing
the square method.
Solution:
Set y = 0. Thus,
x2
+ 4x – 2 = 0
x2
+ 4x = 2
x2
+ 4x + 4 = 2 + 4
(x +2)2
= 6
x + 2 = + 6
x = -2 + 6
The zeros of y = x2
+ 4x -2 are -2 + 6 and -2 - 6 .
Example 3.
Find the zeros of the quadratic function f(x) = x2 + x – 12 using the quadratic
formula.
Solution:
Set y = 0.
In 0 = x2 + x – 12, a = 1, b = 1 and c = -12.

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184
x =
a
acbb
2
42

Use the quadratic formula
x =
)1(2
)12)(1(4)1()1( 2

Substitute the values of a, b and c
x =
2
4811 
Simplify
x =
2
491
x =
2
71
x =
2
71
x =
2
71
x =
2
6
x =
2
8
x = 3 x = -4
The zeros of f(x) = x2 + x – 12 are 3 and -4.
Activity 3 What’s my Rule!
Directions: Work in groups of 3 members each. Perform this activity.
The table below corresponds to a quadratic function. Examine it.
Activity 3. A
a. Plot the points and study the graph. What have you observed?
b. What are the zeros of the quadratic function? How can you identify them?
c. If the zeros are r1 and r2, express the equation of the quadratic function using
f(x) = a(x – r1)(x - r2), where a is any non-zero constant.
d. What is the quadratic equation that corresponds to the table?
Can you think of another way of determining the equation of the quadratic function
from the table of values above?
What if the table of values does not have its zero/s? How can you derive the
equation of the quadratic function?
x -3 -
2
-1 0 1 2 3
y 4 0 -2 -2 0 4 10

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Activity 3.B
The table of values below describes a quadratic function. Find the equation of the
quadratic function by following the given procedure.
a. Substitute 3 ordered pairs (x,y) in y = ax2
+ bx + c
b. What are the three equations you came up with?
___________________, __________________, ___________________
c. Solve for the values of a, b and c.
d. Write the equation of the quadratic function y = ax2
+ bx + c.
How did you obtain the three equations?
What do you call the 3 equations?
How did you solve for the values a, b and c from the three equations?
How can you obtain the equation of a quadratic function from a table of values?
Study the illustrative examples below.
Illustrative example 1
Find a quadratic function whose zeros are -1 and 4.
Solution: If the zeros are -1 and 4, then
x = -1 or x = 4
It follows that
x + 1 = 0 or x - 4 = 0, then
(x+1) (x – 4) = 0
x2 – 3x - 4 = 0
The equation of the quadratic function f(x) = (x2 – 3x – 4) is not unique since
there are other quadratic functions whose zeros are -1 and 4 like f(x) = 2x2 -6x -8,
f(x) = 3x2 – 9x -12 and many more. These equations of quadratic functions are
obtained by multiplying the right-hand side of the equation by a nonzero constant.
Thus, the answer is f(x) = a(x2 – 3x – 4) where a is any nonzero constant.
x -3 -1 1 2 3
y -29 -5 3 1 -5
Did you get the equation of the quadratic function correctly in
the activity? You can go over the illustrative examples below to better
understand the procedure on how to determine the equation of a
quadratic function given the table of values.

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Determine the equation of the quadratic function represented by the table of
values below.
Solution:
Notice that you can’t find any zeros from the given table of values. In this
case, take any three ordered pairs from the table, and use these as the values of x
and y in the equation y = ax2 + bx + c. Let’s say
using point ( 1, 4 ) 4 = a (1)2 + b(1) + c
4 = a + b + c ---------------> equation 1
using point ( -1, 10 ) 10 = a(-1)2 + b(-1) + c
10 = a –b + c ----------------> equation 2
using point ( 2, 4 ) 4 = a(2)2 + b(2) + c
4 = 4a + 2b + c -------------> equation 3
We obtain a system of 3 equations in a, b, and c.
Add corresponding terms in eq.1 and eq. 2 to eliminate b
eq. 1 + eq. 2 4 = a + b + c
10 = a –b + c
We have 14 = 2a + 2c --------------> equation 4
Multiply the terms in eq. 2 by 2 and add corresponding terms in eq. 3 to eliminate b
2(eq .2 ) + eq. 3 20 = 2a – 2b + 2c
4 = 4a + 2b + c
We have 24 = 6a + 3c-------> equation 5
Notice that equation 4 and equation 5 constitute a system of linear equations in two
variables. To solve for c, multiply the terms in equation 4 by 3 and subtract
corresponding terms in equation 5.
3 (eq. 4) – equation 5 42 = 6a + 6c
24 = 6a + 3c
18 = 3c
c = 6
x -3 -2 -1 0 1 2 3
y 24 16 10 6 4 4 6
-

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Substitute the value of c in equation 4 and solve for a.
14 =2a + 2 ( 6)
14 = 2a + 12
2a = 14 – 12
a = 1
Substitute the value of c and a in equation 1 and solve for b.
4 = a + b + c
4 = 1 + b + 6
4 = 7 + b
b = 4 – 7
b = -3
Thus, a = 1, b = -3 and c = 6. Substitute these in f(x) = ax2 + bx + c; the quadratic
function is f(x) = x2 - 3x + 6.
Activity 4 Pattern from curve!
Directions: Work in pairs. Determine the equation of the quadratic function given
the graph by following the steps below.
Study the graph of a quadratic function below.
1. What is the opening of the parabola? What does it imply regarding the value of
a?
2. Identify the coordinates of the vertex.
3. Identify coordinates of any point on the parabola.
4. In the form of quadratic function y = a(x – h)2
+ k, substitute the coordinates of the
point you have taken in the variables x and y and the x- coordinates and y-
coordinate of the vertex in place of h and k, respectively.
5. Solve for the value of a.
6. Get the equation of a quadratic in the quadratic form y = a(x – h)2
+ k.
function by substituting the obtained value of a and the x and y coordinates of the
vertex in h and k respectively.
y
x

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188
When the vertex and any point on the parabola are clearly seen, the equation
of the quadratic function can easily be determined by using the form of a quadratic
function y = a(x – h)2 + k.
Illustrative example 1.
Find the equation of the quadratic function determined from the graph below.
Solution:
The vertex of the graph of the quadratic function is (2, -3). The graph passes
through the point (5, 0). By replacing x and y with 5 and 0, respectively,and h and k
with 2 and -3, respectively, we have
y = a(x- h)2 + k
0 = a(5 – 2)2 -3
0 = a(3)2 -3
3 = 9a
a =
3
1
Thus, the quadratic equation is y =  2
2
3
1
x - 3 or y =
3
5
3
4
3
1 2
 xx .
Aside from the method presented above, you can also determine the equation
of a quadratic function by getting the coordinates of any 3 points lying on the graph.
You can follow the steps in finding the equation of a quadratic function using this
method by following the illustrative example presented previously in this section.
How did you find the activity? Study the mathematical concepts
below to have a clear understanding of deriving the quadratic equation
from the graph.
x
y

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Activity 5 Give my Equation!
Directions: Perform the activity.
A. Study the example below in finding the zeros of the quadratic function and try to
reverse the process to find the solution of the problem indicated in the table on the
right.
Explain the procedure you have done to determine the equation of the quadratic
function.
B. Find the equation of the quadratic function whose zeros are 2 3 .
a. Were you able get the equation of the quadratic function?
b. If no, what difficulties did you encounter?
c. If yes, how did you manipulate the rational expression to obtain the quadratic
function? Explain.
d. What is the equation of the quadratic function?
If r1 and r2 are the zeros of a quadratic function then f(x) = a( x- r1 ) ( x – r2 )
where a is a nonzero constant that can be determined from other point on the graph.
Also, you can use the sum and product of the zeros to find the equation of the
quadratic function. (See the illustrative example in Module 1, lesson 4)
Find the zeros of
f(x) = 6x2
– 7x - 3 using
factoring.
Solution:
f(x) = 6x2
– 7x - 3
0 = 6x2
– 7x – 3
0 = (2x – 3) (3x + 1)
2x – 3 = 0 or 3x + 1 = 0
Then x =
2
3
and x =
3
1

The zeros are
2
3
and
3
1
 .
If the zeros of the quadratic function are 1
and 2, find the equation.
Note: f(x) = a(x- r1) (x – r2) where a is any nonzero constant.
Solution:
_________________________________
_________________________________
_________________________________
_________________________________
_________________________________
_________________________________
Study the mathematical concepts below to have a clearer picture
on how to get the equation of a quadratic function from its zeros.

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190
Example 1
Find an equation of a quadratic function whose zeros are -3 and 2.
Solution
Since the zeros are r1 = -3 and r2 = 2, then
f(x) = a( x- r1 ) ( x – r2 )
f(x) = a[x – (-3)]( x – 2) 
f(x) = a(x + 3)( x – 2 )
f(x) = a(x2 + x – 6) where a is any nonzero constant.
Example 2
Find an equation of a quadratic function with zeros
3
23
.
Solution
A quadratic expression with irrational roots cannot be written as a product of
linear factors with rational coefficients. In this case, we can use another method.
Since the zeros are
3
23
then,
x =
3
23
3x = 23
3x -3 = 2
Square both sides of the equation and simplify. We get
9x2 -18x +9 = 2
9x2 -18x +7 = 0
Thus, an equation of a quadratic function is f(x) = 9x2 -18x +7.
What to PROCESS
Your goal in this section is to apply the concepts you have learned in
finding the zeros of the quadratic function and deriving the equation of
aquadratic function. You will be dealing with some activities and problems to
have mastery of skills needed to perform some tasks ahead.
You learned from the previous activities the methods of finding the zeros of
quadratic function. You also have an initial knowledge of deriving the equation of a
quadratic function from tables of values, graphs,or zeros of the function. The
mathematical concepts that you learned in this section will help you perform the
activities in the next section.

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Activity 6 Match the zeros!
Directions: Matching Type. Each quadratic function has a corresponding letter.
Similarly, each box with the zeros of the quadratic function inside has a
corresponding blank below. Write the indicated letter of the quadratic function on the
corresponding blank below the box containing the zeros of the function to get the
hidden message.
Y f(x) = 4x2 - 25 R f(x) = x2 – 9
V f(x) = 9x2 – 16 E f(x) = x2 -5x – 36
G f(x) = x2 + 6x+ 9 L f(x) = x2 – x – 20
U f(x) = x2 – 4x – 21 D f(x) = 2x2 + x – 3
S f(x) = 6x2 + 5x – 4 O f(x) = 6x2 – 7x + 2
____ ____ ____ ____ ____ ____ ____ ____
____ _____ _____
Activity 7 Derive my equation!
Directions: Work in pairs.
A. Determine the equation of the quadratic function represented by the table of values
below.
B. The vertex of the parabola is ( -3, 5) and it is the minimum point of the graph. If the
graph passes though the point (-2, 7), what is the equation of the quadratic function?
C. Observe the pattern below and draw the 4th
and 5th
figures.
x -4 -3 -2 -1 0 1
y -20 -13 -8 -5 -4 -5






2
1
,
3
2 3,3 






 1,
2
3  4,5 






2
1
,
3
2







3
4
,
3
4  4,9 







2
1
,
3
4







2
5
,
2
5






2
1
,
3
2  3,7 

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192
Make a table of values for the number of squares at the bottom and the total
number of unit squares.
What is the resulting equation of the function?
What method did you use to obtain the equation of the quadratic function in
A? Explain how you obtained your answer.
In B, explain the procedure you used to arrive at your answer. What
mathematical concepts did you apply?
Consider C, did you find the correct equation? Explain the method that you
used to get the answer.
Activity 8 Rule me out!
Directions: Work in pairs to perform this activity. Write the equation in the oval and
write your explanation on the blank provided.
A. Derive the equation of the quadratic function presented by each of the following
graphs below.
1.
2.
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
y
x

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193
3.
4.
5.
Explain briefly the method that you applied in getting the equation
Activity 9 Name the translation!
Directions: Give the equation of a quadratic function whose graphs are described
below. Write your answers on the blanks provided.
1. The graph of f(x) = 3x2
shifted 4 units
downward ______________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
__________________________________________
y
x
y
x
y
x

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194
shifted
2 units to the left ______________________
shifted
5 units upward and 2 units
to the right ______________________
4. The graph of f(x) = -10x2
shifted 2 units
downward and 6 units to the left ______________________
shifted half
unit upward and half unit to the left ______________________
Describe the method you used to formulate the equations of the quadratic functions
above.
Activity 10. Rule my zeros!
Directions: Find one equation for each of the quadratic function given its zeros.
1. 3, 2 _____________________________
2. -2, 5/2 _____________________________
3. 1 + 3 , 1 - 3 _____________________________
4.
3
21
,
3
21
_____________________________
5.
3
11
, -
3
11
_____________________________
In your reflection note, explain briefly the procedure used to get the equation of the
quadratic function given its zero/s.

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195
Activity 11 Dare to hit me!
Directions: Work in pairs. Solve the problem below:
Problem. The path of the golf ball follows a trajectory. It hits the ground 400 meters
away from the starting position. It just overshoots a tree which is 20 m high and is
300m away from the starting point.
From the given information, find the equation determined by the path of the golf ball.
Your goal in this section is to have a deeper understanding on how to
derive the equation of the quadratic functions. You can apply the skills you have
learned from previous sections to perform the tasks ahead. The activities
provided for you in this section will be of great help to deepen your understanding
for further application of the concepts.
Did you enjoy the activities in this section?
Go back to the previous section and compare your initial ideas with
the discussion. How much of your initial ideas are found in the discussion?
Which ideas are different and need to be clarified more?

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Activity 12 Connect and Relate!
Directions: Work in groups of 5 members. Perform this mathematical investigation.
Joining any two points on a circle determine a chord. A chord is a line
segment that joins any two points on a circle. Investigate the relationship between
the number of points n on the circle and the maximum number of chords C that can
be drawn. Make a table of values and find the equation of the function representing
the relationship. How many chords are there if there are 50 points on a circle?
What kind of function is represented by the relationship?
Given the number of points, how can you easily determine the maximum number of
chords that can be drawn?
How did you get the pattern or the relationship?
Activity 13 Profit or Loss!
Directions: Work in pairs. Analyze the graph below and answer the questions that
follow.
a. Describe the graph.
b. What is the vertex of the graph? What does the vertex represent?
c. How many weeks should the owner of the banana plantation wait before harvesting
the bananas to get the maximum profitmmmmmmmmm ?
d. What is the equation of the function?
P
w

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Activity 14 What if questions!
Directions: Work in groups with 3 members each.
1. Given the zeros r1 and r2 of the following quadratic function, the equation of the
quadratic function is f(x) = a(x - r1)(x - r2) where a is any nonzero constant. Consider
a = 1 in any of the situations in this activity.
a. -2 and 3.
b. 3 3
A. What is the equation of the quadratic function?
a. ____________________
b. ____________________
B. If we double the zeros, what is the new equation of the quadratic function?
a. _____________________
b. _____________________
Find a pattern on how to determine easily the equation of a quadratic
function given this kind of condition.
C. What if the zeros are reciprocal of the zeros of the given function? What is the new
equation of the quadratic function?
a. _____________________
b. _____________________
Find the pattern.
D. What if you square the zeros? What is the new equation of the quadratic function?
a. _____________________
b. _____________________
Find the pattern.
2. Find the equation of the quadratic function whose zeros are
a. the squares of zeros of f(x) = x2
– 3x -5.
b. the reciprocal of the zeros of f(x) = x2
– x -6
c. twice the zeros of f(x) = 3x2
– 4x -5

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Activity 15 Principle Pattern Organizer!
Directions: Make a summary of what you have learned.
What to TRANSFER
Did you enjoy the activities? I hope that you learned a lot in this section
and you are now ready to apply the mathematical concepts you learned in all the
activities and discussions from the previous sections.
In this section, you will be given a task wherein you will apply what you
have learned in the previous sections. Your performance and output will show
evidence of your learning.
Deriving quadratic function from:
Graphs
Zero
s
Table of Values

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http://64.19.142.13/australia.gov.au/sites/default/files
/agencies/culture/library/images/site_images/bridges
unset-web.jpg
Activity 16 Mathematics in Parabolic Bridges!
GOAL: Look for world famous parabolic bridges and determine the equation of
the quadratic functions .
ROLE: Researchers
AUDIENCE: Head of the Mathematics Department and math teachers
SITUATION: For the Mathematics monthly
culminating activity, your section is tasked to
present a simple research paper in
Mathematics. Your group is assigned to make a
simple research on the world’s famous parabolic
bridges and the mathematical
equations/functions described by each bridge.
Make a simple research on parabolic
bridges and use the data to formulate the
equations of quadratic functions pertaining to
each bridge.
PRODUCT: Simple research paper on world famous parabolic bridge.
STANDARDS FOR ASSESSMENT:
You will be graded based on the rubric designed suitable for your task and
performance.

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Lesson 4. Applications of Quadratic Functions
What to KNOW
Activity 1
Directions: Consider this problem.
1. If the perimeter of the rectangle is 100 m, find its dimensions if its area is maximum.
a. Complete the table below for the possible dimensions of the rectangle and their
corresponding areas. The first column has been completed for you.
b. What is the largest area that you obtained?
c. What are the dimensions of a rectangle with the largest area?
d. The perimeter P of the given rectangle is 100. Make a mathematical statement for
the perimeter of the rectangle.
e. Simplify the obtained equation and solve for the length l of the rectangle in terms
of its width w.
f. Express the area A of a rectangle as a function of width w.
g. What kind of equation is the result?
h. Express the function in standard form. What is the vertex?
i. Graph the data from the table in a showing the relationship between the width
and the area.
j. What have you observed about the vertex of the graph in relation to the
dimensions and the largest area?
Width (w) 5 10 15 20 25 30 35 40 45 50
Length (l) 45
Area (A) 225
The application of quadratic function can be seen in many
different fields like physics, industry, business and in variety of
mathematical problems. In this section, you will explore situations that
can be modeled by quadratic functions.
Let us begin this lesson by recalling the properties of quadratic
functions and how we can apply them to solve real life problems.
How did you find the activity? Can you still recall the properties of
quadratic function? Did you use them in solving the given problem?
To better understand how the concepts of the quadratic function
can be applied to solve geometry problems, study the illustrative example
presented below.

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Example 1.
What are the dimensions of the largest rectangular field that can be enclosed by 80
m of fencing wire?
Solution:
Let l and w be the length and width of a rectangle. Then, the perimeter P of a
rectangle is P = 2l + 2w. Since P = 80 m, thus,
2l + 2w = 80
l + w = 40 It follows that
l = 40 – w expressing the length as a function of w
l = 40 – w Substituting in the formula for the area A of a
rectangle
A(w) = wl
A(w) = w ( 40 – w )
A(w) = -w2 + 40w
By completing the square,
A(w) = - ( w - 20 )2 + 400
The vertex the graph of the function A(w) is (20, 400). This point indicates the a
maximum value of 400 for A(w) that occurs when w = 20. Thus, the maximum area is 400 m2
when the width is 20. If the width is 20 m, then the length is (40 – 20) m or 20 m also. The
field with maximum area is a square.
Activity 2 Catch me when I fall!
Directions: Work in groups with 3 members each. Do the following activity.
Problem: The height (H) of the ball thrown into the air with an initial velocity of 9.8 m/s from
a height of 2 m above the ground is given by the equation H(t) = -4.9t2
+ 9.8t + 2, where t is
the time in seconds that the ball has been in the air.
a. What maximum height did the object reach?
b. How long will it take the ball to reach the maximum height?
c. After how many seconds is the ball at a height of 4 m?
Guide questions:
1. What kind of function is the equation H(t) = -4.9t2
+ 9.8t + 2?
2. Transform the equation into the standard form.
3. What is the vertex?
4. What is the maximum height reached by the ball?
w

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202
5. How long will it take the ball to reach the maximum height?
6. If the height of the ball is 4 m, what is the resulting equation?
7. Find the value of t to determine the time it takes the ball to reach 4 m .
Free falling objects can be modeled by a quadratic function h(t) = –4.9t2 + V0t
+ h0, where h(t) is the height of an object at t seconds, when it is thrown with an
initial velocity of V0 m/s and an initial height of h0 meters. If units are in feet, then the
function is h(t) = –16t2 + V0t + h0.
From a 96-foot building, an object is thrown straight up into the air then
follows a trajectory. The height S(t) of the ball above the building after t seconds is
given by the function S(t) = 80t – 16t2.
1. What maximum height will the object reach?
2. How long will it take the object to reach the maximum height?
3. Find the time at which the object is on the ground.
Solution
1. The maximum height reached by the object is the ordinate of vertex of the parabola of
the function S(t) = 80t – 16t2
. By transforming this equation in completed square form we
have,
S(t) = 80t – 16t2
S(t) = – 16t2 + 80t
S(t) = – 16(t2
- 5t)
S(t) = – 16(t2
- 5t +
4
25
) + 100
S(t) = – 16(t -
2
5
)2
+ 100
The vertex is (
2
5
, 100) Thus the maximum height reached by the object is 100
ft from the top of the building. This is 196 ft from the ground.
2. The time for an object to reach the maximum height is the abscissa of the vertex
of the parabola or the value of h.
How did you find the preceding activity? The previous activity
allowed you to recall your understanding of the properties of quadratic
function and gave you an opportunity to solve real-life related problems that
deal with quadratic function
The illustrative example below is intended for you to better
understand the key ideas necessary to solve real-life problems involving
quadratic function.

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203
S(t) = 80t – 16t2
S(t) = – 16(t -
2
5
)2
+ 100
Since the value of h is
2
5
or 2.5, then the object is at its maximum height after
2.5 seconds.
3. To find the time it will take the object to hit the ground, let S(t) = -96 , since the
height of the building is 96 ft. The problem requires us to solve for t.
h(t) = 80t – 16t2
-96= 80t – 16t2
16t2 -80t -96 = 0
t2 – 5t - 6 = 0
( t -6)( t + 1) = 0
t = 6 or t = -1
Thus, it will take 6 seconds before the object hits the ground.
Activity 3 Harvesting Time!
Directions: Solve the problem by following the given steps.
Problem: Marvin has a mango plantation. If he picks the mangoes now, he will get 40
small crates and make a profit of P 100 per crate. For every week that he delays
picking, his harvest increases by 5 crates. But the selling price decreases by P10 per
crate. When should Marvin harvest his mangoes for him to have the maximum
profit?
a. Complete the following table of values.
b. Plot the points and draw the graph of the function.
c. How did you determine the total profit?
d. Express the profit P as a function of the number of weeks of waiting.
e. Based on the table of values and graph, how many weeks should Marvin wait
before picking the mangoes to get the maximum profit?
This problem is adapted from PASMEP Teaching Resource Materials, Volume II.
No. of weeks of waiting
(w)
0 1
No. of crates 40
Profit per crates (P) 100
Total profit (T)

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204
Illustrative example
Problem. A garments store sells about 40 t-shits per week at a price of Php100 each.
For each P10 decrease in price, the sales lady found out that 5 more t-shits per
week were sold.
a. Write a quadratic function in standard form that models the revenue from t-shirt
sales.
b. What price produces the maximum revenue?
Solution:
You know that Revenue R(x) = (price per unit) x (number of units produced or sold).
Therefore, Revenue R(x) = (Number of t-shirts sold) (Price per t-shirt)
Revenue R(x) = (40 + 5x) (100-10x)
R(x) = -50x2 + 100x +4000
If we transform the function into the form y = a(x- h)2 + k
R(x) = -50(x -1)2 + 4050
The vertex is (1, 4050).
Thus, the maximum revenue is Php 4050
The price of the t-shirt to produce maximum revenue can be determined by
P(x) = 100 – 10x
P(x) = 100 – 10 (1) = 90
Thus, Php 90 is the price of the t-shirt that produces maximum revenue.
How did you find the activity? A quadratic function can be applied
in business/industry to determine the maximum profit, the break-even
situation and the like.
Suppose x denotes the number of units a company plans to
produce or sell. The revenue function R(x) is defined as R(x)= (price per
unit) x (number of units produced or sold).
Study the example below.

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205
What to PROCESS
Activity 4 Hit the mark!
Directions: Analyze and solve this problem.
Problem A.
A company of cellular phones can sell 200 units per month at P 2 000 each.
Then they found out that they can sell 50 more cell phone units every month for each
P 100 decrease in price.
a. How much is the sales amount if cell phone units are priced at P2000 each?
b. How much would be their sales if they sell each cell phone unit at P 1600?
c. Write an equation for the revenue function?.
d. What price per cell phone unit gives them the maximum monthly sales?
e. How much is the maximum sale?
Problem B
The ticket to a film showing costs P 20. At this price, the organizer found out
that all the 300 seats are filled. The organizer estimates that if the price is increased,
the number of viewers will fall by 50 for every P 5 increase.
a. What ticket price results in the greatest revenue?
b. At this price, what is the maximum revenue?
What properties of a quadratic function did you use to come up with the correct
solution to the problem A? problem B?
Activity 5. Equal Border!
Directions: Work in pairs and perform this activity.
A photograph is 16 inches wide and 9 inches long and is surrounded by a frame of
uniform width x. If the area of the frame is 84 square inches, find the uniform width of the
frame.
a. Make an illustration of the described photograph.
b. What is the area of the picture?
c. If the width of the frame is x inches, what is the length and width of the
photograph and frame?
d. What is the area of the photograph and frame?
e. Given the area of the frame which is 84 square inches, formulate the relationship
among three areas and simplify.
f. What kind of equation is formed?
g. How can you solve the value of x?
h. How did you find the activity? What characteristics of quadratic functions did you
apply to solve the previous problem?
Your goal in this section is to extend your understanding and skill in
the use of quadratic function to solve real-life problems.

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206
Activity 6 Try this!
Directions: With your partner, solve this problem. Show your solution.
A. An object is thrown vertically upward with a velocity of 96 m/sec. The distance S(t)
above the ground after t seconds is given by the formula S(t) = 96t – 5t2
.
a. How high will it be at the end of 3 seconds?
b. How much time will it take the object to be 172 m above the ground?
c. How long will it take the object to reach the ground?
B. Suppose there are 20 persons in a birthday party. How many handshakes are there
altogether if everyone shakes hands with each other?
a. Make a table of values for the number of persons and the number of
handshakes.
b. What is the equation of the function?
c. How did you get the equation?
d. If there are 100 persons, how many handshakes are there given the same
condition?
In problem A, what mathematical concepts did you apply to solve the problem?
If the object reaches the ground, what does it imply?
In problem B, what are the steps you follow to arrive at your final answer?
Did you find any pattern to answer the question d?
Your goal in this section is to have a deeper understanding on how
to solve problems involving quadratic functions. The activities provided for
you in this section will be of great help to practice the key ideas developed
throughout the lesson and to stimulate your synthesis of the key principles
and techniques in solving problems on quadratic functions.
Did you enjoy the activities in this section? Go back to the previous
section and compare your initial ideas with the discussion. How much of
your initial ideas are found in the discussion? Which ideas are different and
need revision?

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207
Activity 7 Geometry and Number!
Directions: Solve the problems. Show your solution.
1. What are the dimensions of the largest rectangular field that can be enclosed with 60
m of wire?
2. Find the maximum rectangular area that can be enclosed by a fence that is 364
meters long.
3. Find two numbers whose sum is 36 and whose product is a maximum.
4. The sum of two numbers is 28. Find the two numbers such that the sum of their
squares is a minimum?
5. Marlon wants to fence a rectangular area that has one side bordered by an irrigation.
If he has 80 m of fencing materials, what are the dimensions and the maximum area
he can enclose?
6. The length of a rectangular field is 8 m longer than its width If the area is 2900m2
,
find the dimensions of the lot.
7. The sum of two numbers is 24. Find the numbers if their product is to be a maximum.
Activity 8 It’s high time!
Directions: Work in group with 5 members each. Solve the problems. Show your
solution.
1. A ball is launched upward at 14 m/s from a platform that is 30 m high.
a. Find the maximum height the ball reaches.
b. How long it will take the ball to reach the maximum height?
c. How long will it take the ball to reach the ground?
2. On top of a hill, a rocket is launched from a distance 80 feet above a lake. The rocket
will fall into the lake after its engine burns out. The rocket’s height h, in feet above the
surface of the lake is given by the equation h = -16t2
+ 64t +80, where t
is time in seconds. What is the maximum height reached by the rocket?
3. A ball is launched upward at 48 ft/s from a platform that is 100 ft. high. Find the
maximum height the ball reaches and how long it will take to get there.
4. An object is fired vertically from the top of a tower. The tower is 60.96 ft high. The
height of the object above the ground t seconds after firing is given by the formula
h(t) = -16t2
+ 80t + 200. What is the maximum height reached by the object? How
long after firing does it reach the maximum height?
5. The height in meters of a projectile after t seconds is given by h(t) = 160t – 80t2
. Find
the maximum height that can be reached by the projectile.
6. Suppose a basketball is thrown 8 ft from the ground. If the ball reaches the 10m
basket at 2.5 seconds, what is the initial velocity of the basketball?

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208
Activity 9 Reach the target!
Directions: Work in pairs to solve the problems below. Show your solution.
1. A store sells lecture notes, and the monthly revenue R of this store can be modelled
by the function R(x) = 3000 +500x -100x2
, where x is the peso increase over Php 4.
What is the maximum revenue?
2. A convention hall has a seating capacity of 2000. When the ticket price in the concert
is Php160, attendance is 500. For each Php 20 decrease in price, attendance
increases by 100.
a. Write the revenue R of the theater as a function of concert ticket price x.
b. What ticket price will yield maximum revenue?
c. What is the maximum revenue?
3. A smart company has 500 customers paying P 600 each month. If each Php 30
decrease in price attracts 120 additional customers, find the approximate price that
yields maximum revenue?
Activity 10 Angles Count!
Directions: Work in groups of 5 members each. Perform this mathematical
Investigation.
Problem. An angle is the union of two noncollinear rays. If there are 100 rays in the
figure, how many angles are there?
a. What kind of function is represented by the relationship?
b. Given the number of rays, how can you determine the number of angles?
c. How did you get the pattern or the relationship?
What to TRANSFER
In working on problems and exploration in this section, you
studied the key ideas and principles to solve problems involving quadratic
functions. These concepts will be used in the next activity which will
require you to illustrate a real-life application of a quadratic function.
In this section, you will be given a task wherein you will apply
what you have learned in the previous sections. Your performance and
output in the activity must show evidence of your learning.

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209
Activity 11 Fund Raising Project!
GOAL: Apply quadratic concepts to plan and organize a fund raising activity
ROLE: Organizers of the Event
SITUATION: The Mathematics Club plan to sponsor a film viewing on the last Friday of the
Mathematics month. The primary goal for this film viewing is to raise funds for their Math
Park Project and of course to enhance the interest of the students in Mathematics.
To ensure that the film viewing activity will not lose money, careful planning is
needed to guarantee a profit for the project. As officers of the club, your group is tasked to
make a plan for the event. Ms. De Guzman advised you to consider the following variables in
making the plan.
a) Factors affecting the number of tickets sold
b) Expenses that will reduce profit from ticket sales such as:
- promoting expenses
- operating expenses
c) How will the expenses depend on the number of people who buy tickets
and attend?
d) Predicted income and ticket price
e) Maximum income and ticket price
f) Maximum participation regardless of the profit
g) What is the ticket price for which the income is equal to the expenses?
Make a proposed plan for the fund raising activity showing the relationship of the
related variables and the predicted income, price, maximum profit, maximum participation,
and also the break-even point.
AUDIENCE: Math Club Advisers, Department Head-Mathematics,
Mathematics teachers
PRODUCT: Proposed plan for the fund raising activity (Film showing)
STANDARD: Product/Performance will be assessed using a rubric.
Summary/Synthesis/Feedback
This module was about concepts of quadratic functions. In this module, you were
encouraged to discover by yourself the properties and characteristics of quadratic functions.
The knowledge and skills gained in this module help you solve real-life problems involving
quadratic functions which would lead you to perform practical tasks. Moreover, concepts you
learned in this module allow you to formulate real-life problems and solve them in a variety of
ways.

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210
Glossary of Terms
axis of symmetry – the vertical line through the vertex that divides the parabola into
two equal parts
direction of opening of a parabola – can be determined from the value of a in
f(x) = ax2 + bx + c. If a > 0, the parabola opens upward; if a < 0, the parabola opens
downward.
domain of a quadratic function – the set of all possible values of x. Thus, the
domain is the set of all real numbers.
maximum value – the maximum value of f(x) = ax2 + bx + c where a < 0, is the y-
coordinate of the vertex.
minimum value – the minimum value of f(x) = ax2 + bx + c where a > 0, is the y-
coordinate of the vertex.
parabola – the graph of a quadratic function.
quadratic function- a second-degree function of the form f(x) = ax2 + bx + c, where
a, b, and c are real numbers and a 0 . This is a function which describes a
polynomial of degree 2.
Range of quadratic function - consists of all y greater than or equal to the y-
coordinate of the vertex if the parabola opens upward.
- consists of all y less than or equal to the y-coordinate of the vertex if the parabola
opens downward
vertex – the turning point of the parabola or the lowest or highest point of the
parabola. If the quadratic function is expressed in standard form y = a(x – h)2 + k, the
vertex is the point (h, k).
zeros of a quadratic function – the x-intercepts of the parabola.

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211
REFERENCES
Basic Education Curriculum (2002)
Basic Education Assistance for Mindanao. (2008) Module 3:Quadratic Functions and
their Graphs (Learning Guide 6), p. 14-15, 34, 37, 40-41, 44
Catao, E. et. Al. PASMEP Teaching Resource Materials, Volume II
Cramer, K., (2001) Using Models to Build Middle-Grade Students' Understanding of
Functions. Mathematics Teaching in the Middle School. 6 (5),
De Leon, Cecile, Bernabe, Julieta. (2002) Elementary Algebra. JTW Corporation,
Quezon City, Philippines.
Hayden, J., & Hall, B. (1995) Trigonometry, (Philippine Edition) Anvil Publishing Inc.,
Quezon City, Philippines.
Gallos, F., Landrito, M. & Ulep, S. (2003) Practical Work Approach in High School
Mathematics (Sourcebook fro Teachers), National Institute for Science and
Mathematics Education Development, Diliman, Quezon City.
Hernandez, D. et al. (1979) Functions (Mathematics for Fourth Year High School),
Ministry of Education, Culture and Sports. Capitol Pub. House Inc., Diliman
Quezon City.
INTEL, Assessment in the 21st Century Classroom E Learning Resources.
Joson, Lolita, Ymas Jr., Sergio. (2004) College Algebra. Ymas Publishing House,
Manila, Philippines.
Lapinid, M., & Buzon, O. (2007) Advanced Algebra, Trigonometry and
Statistics,Salesiana Books by Don Bosco Press, Inc., Makati City.
Leithold, L. (1992). College Algebra and Trigonometry. Philippines :National
Book Store Inc.
Marasigan, J., Coronel, A. et. Al. Advanced Algebra with Trigonometry and Statistics,
The Bookmark Inc., San Antonio Villagr, Makati City.
Numidos, L. (1983) Basic Algebra for Secondary Schools, Phoenix Publishing House
Inc., Quezon Avenue, Quezon City.
Orines, Fernando B., (2004). Advanced Algebra with Trigonometry and Statistics:
Phoenix Publishing House, Inc.

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212
WEBLINKS
Website links for Learning Activities
1. http://www.analyzemath.com/quadraticg/Problems1.html
2. http://www.learner.org/workshops/algebra/workshop4/
3. http://www.cehd.umn.edu/ci/rationalnumberproject/01_1.html
4. http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U
10_L2_T1_text_container.html
5. http://answers.yahoo.com/question/index?qid=20090531221500AAoXyOi
6. http://algebra2trigplt.wikispaces.com/file/view/cb_sb_math_miu_L6_U3_embedded_a
ssessment3-1_SE_v2.pdf
7. http://www.youtube.com/watch?v=BYMd-7Ng9Y8
8. http://www.utdanacenter.org/mathtoolkit/downloads/scope/OLDalg2scope/ovadawatt
a.pdf
9. http://www.westfargo.k12.nd.us/district/academic/images/MathProblemSolvingRubric
.pdf
10. http://www.mathopenref.com/quadraticexplorer.html
11. http://mathitude.perso.sfr.fr/PDF/safety_kit_1.pdf
12. http://mathwithclayton.wikispaces.com/file/view/unit07.pdf/418858436/unit07.pdf
13. http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U10_L2_T1
_text_container.html
14. http://www.teacherweb.com/ny/arlington/algebraproject/U5L6ApplicationsofQuadraticFun
ctionsDay1.pdf
15. http://www.youtube.com/watch?v=5bKch8vitu0
16. http://ts2.mm.bing.net/th?id=H.4577714805345361&pid=15.1
17. http://www.mathopenref.com/quadraticexplorer.html
18. http://fit-ed.org/intelteachelements/assessment/index.php
Website links for Images
1. http://www.popularmechanics.com/cm/popularmechanics/images/y6/free-throw-0312-
mdn.jpg
2. http://sites.davidson.edu/mathmovement/wp
content/uploads/2011/11/trajectories.png
3. http://web.mnstate.edu/lindaas/phys160/lab/Sims/projectileMotion.gif -throw-0312-
mdn.jpg
4. http://64.19.142.13/australia.gov.au/sites/default/files/agencies/culture/library/images/
site_images/bridgesunset-web.jpg

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