Solving quadratic equations

Equations and Quadratic
functions
Objective
Solving quadratic equation by
Factorisation

Warm up
• Solve the given equation.
 
 
y
y
y
y
y



3
6
1
2
6
3
11 5
3
5 11
3

11.6 – Solving Quadratic Equations by Factoring
A quadratic equation is written in the Standard
Form, ax2 bx  c  0
where a, b, and c are real numbers and . 0 a 
Examples:
2 x 7x 12  0
xx 7  0
2 3x  4x 15
(standard form)

Zero Factor Property:
If a and b are real numbers and if , 0 ab 
then or . 0 a  0 b 
Examples:
xx 7  0
x  0 x 7  0
x  0 x  7

Zero Factor Property:
If a and b are real numbers and if , 0 ab 
then or . 0 a  0 b 
Examples:
x103x6  0
x 10  0 3x6  0
x 1010  010 3x 66  06
x 10
x

3 6
3 3
3x  6 x  2

Solving Quadratic Equations:
1) Write the equation in standard form.
2) Factor the equation completely.
3) Set each factor equal to 0.
4) Solve each equation.
5) Check the solutions (in original equation).

2 x 3x 18
x2 3x 18  0
Factors of 18:
1, 18 2, 9 3, 6
x3
x 3  0 x 6  0
x  3 x  6
    2
6 3 6 18
3618 18
18 18 
    2
3 3 3 18
99 18
18 18
  6 x 0

If the Zero Factor
Property is not used,
then the solutions will
be incorrect
2 x 3x 18
xx3 18
x 18
    2
18 3 18 18
32454 18
270 18 
    2
21 3 21 18
4416318
378 18
x318
x33183
x  21

xx 4  5
x2  4x  5
2 x  4x 5  0
x 1x 5  0
x1 0 x 5  0
x  1 x  5

x 33x 2  0
3x2  7x  6 30x 3 2 0 x 
3 x 
2
3
x 
x3x 7  6
2 3 7 6 0 x x    32x 
Factors of 3:
1, 3
Factors of 6:
1, 6 2, 3

9x2  24x  16
2 9x 24x 16  0
9 and 16 are perfect squares
3x 43x 4  0
3x4  0
3x  4
4
3
x 

2x3 18x  0
2x
2x
  2 9 x  0 
  3 x    3 x  0 
x 3  0 x3 0
2x  0
x  0 x  3
x  3

x 33x2 20x 7  0
3: Factors of 1, 37 : Factors of 1, 7
x3
30 x 
  7 x  0   3 1 x 
70x 3 1 0 x 
x  3 x  7
3x  1
1
3
x  

Continuous assessment
• Q1: Solve these equations by factorisation.
a) (x-2)(2x+1) = 0
b) 3x2 – 27x = 0
c) 2x2 – 7x + 6 = 0
Q2: a) 2(d2 – 3d +3) = d + 1
b) 3(e + 1)2= 1 – e
c) (g + 3)(2 – g) = g2

Answers
• Q1: a) x = 2, x = - ½ b) x = 0, x= 9 c) x = -4, x=3
• Q2: a) (a+3)(a-3) b) e = -2, e = -1/3
c) d = -2, d = 1 ½

Final assessment
• Solve by factorisation.
17x2 – 51x + 34 = 0

11.7 – Quadratic Equations and Problem Solving
A cliff diver is 64 feet above the surface of the
water. The formula for calculating the height (h)
of the diver after t seconds is: h  16t2 64.
How long does it take for the diver to hit the
surface of the water?
2 16t  64
16   2 t  4
16   2 t    2 t 
0 
0 
0 
t 2  0 t 2  0
t  2 t  2 seconds

The square of a number minus twice the number
is 63. Find the number.
x is the number.
2 x
2x  63 
2 x  2x 63  0
63: Factors of 1, 63 3, 21 7, 9
 x  7
  9 x 0
x 7  0 x 9  0
x  7
x  9

The length of a rectangular garden is 5 feet more than
its width. The area of the garden is 176 square feet.
What are the length and the width of the garden?
l wA
The width is w.
The length is w+5. w5w176
  11 w 
w11 0
w11
2 w 5w 176
2 w 5w176  0
w16  0
w 16
w11 l 115
l 16
feet
feet
Factors of 176:
1,176 2, 88 4, 44
8, 22 11,16
  16 w  0 

Find two consecutive odd numbers whose product is
23 more than their sum?
Consecutive odd numbers: x
x
2. x 
5x 5x 2 2 2 25 x x x   
2 x  25
2 x  25  0
x5
50 x  50 x 
52  3 5 2  7
5, 3 5, 7
  2 x    2 x x   23
2 x  2x  2x  2x  25 2x
2 x  25  25 25
x5  0

The length of one leg of a right triangle is 7 meters less than
the length of the other leg. The length of the hypotenuse is
13 meters. What are the lengths of the legs?
ax
a 12
Pythagorean Th.
 2 2 2 x  x 7 13
x  5
meters
 5
7bx 13 c 
2 2 x  x 14x  49 169
2 2x 14x 120  0
  2 2 x 7x 60  0
2
50x 12 0 x 
x 12
b 127 meters
2 2 2 a b  c
Factors of 60:1, 60 2, 30
3, 20 4,15 5,12
  5 x   12 x  0
6,10

Solving quadratic equations

More Related Content

What's hot (20)

Similar to Solving quadratic equations (20)

Solving quadratic equations