Lesson 4.3 regular
- 1. 4.3 Solving Quadratic Equations by Factoring A quadratic equation is written in the Standard Form, 2 ax + bx + c = 0 where a, b, and c are real numbers and a ≠ 0. Examples: x − 7 x + 12 = 0 x ( x + 7) = 0 2 3x + 4 x = 15 2 (standard form)
- 2. 14.3 Solving Quadratic Equations by Factoring Zero Factor Property: If a and b are real numbers and if ab = 0 , then a = 0 or b = 0 . Examples: x ( x + 7) = 0 x=0 x+7 =0 x=0 x = −7
- 3. 4.3 Solving Quadratic Equations by Factoring Zero Factor Property: If a and b are real numbers and if ab = 0 , then a = 0 or b = 0 . Examples: ( x − 10 ) ( 3x − 6 ) = 0 x − 10 = 0 3x − 6 = 0 x − 10 + 10 = 0 + 10 3 x − 6 + 6 = 0 + 6 3x 6 x = 10 = x=2 3x = 6 3 3
- 4. 11.6 – Solving Quadratic Equations by Factoring 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).
- 5. 11.6 – Solving Quadratic Equations by Factoring x − 3 x = 18 2 x − 3x − 18 = 0 Factors of 18 : 1, 18 2, 9 3, 6 ( 6) ( x + 3) ( x − 6 ) ( −3) 2 x+3= 0 x = −3 =0 x−6 = 0 x=6 2 − 3 ( 6 ) = 18 36 − 18 = 18 18 = 18 2 − 3 ( −3) = 18 9 + 9 = 18 18 = 18
- 6. 4.3 Solving Quadratic Equations by Factoring If the Zero Factor Property is not used, then the solutions will be incorrect x − 3 x = 18 x ( x − 3) = 18 x = 18 x − 3 = 18 ( 18) 2 324 − 54 = 18 270 ≠ 18 2 x − 3 + 3 = 18 + 3 x = 21 − 3 ( 1 8 ) = 18 ( 21) 2 − 3 ( 21) = 18 441 − 63 = 18 378 ≠ 18
- 7. 4.3 Solving Quadratic Equations by Factoring x ( x − 4) = 5 x − 4x = 5 2 x − 4x − 5 = 0 2 ( x + 1) ( x − 5) = 0 x +1 = 0 x −5 = 0 x = −1 x=5
- 8. 4.3 Solving Quadratic Equations by Factoring x ( 3x + 7 ) = 6 ( x + 3) ( 3 x − 2 ) = 0 3x + 7 x = 6 x + 3 = 0 3x − 2 = 0 x = −3 3x = 2 x=2 3 2 3x + 7 x − 6 = 0 Factors of 3 : 1, 3 Factors of 6 : 1, 6 2, 3 2
- 9. 11.6 – Solving Quadratic Equations by Factoring 9 x − 24 x = −16 2 9 x − 24 x + 16 = 0 2 ( 9 and 16 are perfect squares ) ( 3x − 4 ) ( 3x − 4 ) = 0 3x − 4 = 0 3x = 4 4 x= 3
- 10. 11.6 – Solving Quadratic Equations by Factoring 2 x − 18 x = 0 2 2x ( x − 9 ) = 0 3 2x ( x + 3) ( x − 3) = 0 2x = 0 x + 3 = 0 x − 3 = 0 x=3 x=0 x = −3
- 11. 11.6 – Solving Quadratic Equations by Factoring ( x + 3) ( 3 x − 20 x − 7 ) = 0 Factors of 3 : 1, 3 Factors of 7 : 1, 7 2 ( x + 3) ( x − 7 ) ( 3x + 1) = 0 x+3= 0 x = −3 x−7 = 0 x=7 3x + 1 = 0 3 x = −1 1 x=− 3
- 12. 4.3 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 = −16t 2 + 64. How long does it take for the diver to hit the surface of the water? 2 0 = −16t + 64 2 0 = −16 ( t − 4 ) 0 = −16 ( t + 2 ) ( t − 2 ) t+2=0 t = −2 t−2=0 t = 2 seconds
- 13. 4.3 Quadratic Equations and Problem Solving The square of a number minus twice the number is 63. Find the number. x is the number. x −2x = 63 2 x − 2 x − 63 = 0 Factors of 63 : 1, 63 3, 21 7, 9 2 ( x + 7 ) ( x − 9) =0 x+7 =0 x−9 = 0 x = −7 x=9
- 14. 11.7 – Quadratic Equations and Problem Solving 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 ×w = A The width is w. The length is w+5. ( w + 5) w = 176 ( w − 11) ( w + 16 ) w + 5w = 176 w − 11 = 0 w = 11 2 w2 + 5w − 176 = 0 Factors of 176 : 1, 176 2, 88 4, 44 8, 22 11, 16 w = 11 feet =0 w + 16 = 0 w = −16 l = 11 + 5 l = 16 feet
- 15. 4.3 Quadratic Equations and Problem Solving Find two consecutive odd numbers whose product is 23 more than their sum? x + 2. Consecutive odd numbers: x x ( x + 2 ) = ( x + x + 2 ) +23 x + 5 = 0 2 x + 2 x = 2 x + 25 x = −5 x 2 + 2 x − 2 x = 2 x + 25 − 2 x −5 + 2 = −3 2 x = 25 −5, − 3 2 x − 25 = 25 − 25 x 2 − 25 = 0 ( x + 5) ( x − 5) = 0 x −5 = 0 x=5 5+ 2 = 7 5, 7
- 16. 4.3 Quadratic Equations and Problem Solving 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? ( Pythagorean Th.) a 2 + b2 = c 2 a = x b = x − 7 c = 13 2 ( x + 5 ) ( x − 12 ) = 0 2 2 x + ( x − 7 ) = 132 x − 12 = 0 x+5 = 0 x 2 + x 2 − 14 x + 49 = 169 x = −5 x = 12 2 x 2 − 14 x − 120 = 0 a = 12 meters 2 2 ( x − 7 x − 60 ) = 0 b = 12 − 7 = 5 meters Factors of 60 : 1, 60 2, 30 3, 20 4, 15 5, 12 6, 10