A quadratic expression is a mathematical expression that can be written in the form 2

Transcription

1 118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is a mathematical expression that can be written in the form ax + bx + c, where a 0. A quadratic equation is a second-degree equation. In other words, the highest exponent on the variable is two. The standard form of a quadratic equation is Not all quadratics are factorable. ax + bx + c 0. Factoring Quadratics Using the ac-method. 1. Write the quadratic in standard form.. Determine the product ac.. Find two numbers, d and e, that multiply to form the product ac and sum to b.. If the two numbers do not exist, the equation is not factorable. If the two numbers do exist, rewrite the quadratic as ax + dx + ex + c. 5. Proceed as in factoring by grouping. a. Make two pairs: ax + dx and ex + c. b. Factor each pair. c. Factor out the greatest common factor (GCF). Example Solution a) Factor: x x Identify a, b, and c. a 1, b -, c -15. Multiply a and c. ac (1)(-15) -15. Find two numbers whose product is 15 and sum to. Factors Product Sum Conclusion -1, No 1, No -, No, Yes. Rewrite. x x 15 x + x 5x Factor by grouping: x + x 5x 15 (x + x) + ( 5x 15) x(x + ) 5(x + ) (x + )(x 5) Answer: x x 15 (x + )(x 5) Note: Factoring results can be checked by multiplying the factors. Check: (x + )(x 5) x 5x + x 15 x x 15

2 SECTION.6 Factoring and the Quadratic Equation 119 Example Solution b) Factor: x 8x + 1. Identify a, b, and c. a, b -8, c. Multiply a and c. ac ()() 1. Find two numbers whose product is 1 and sum to 8. Check Your Progress.6 Factor each of the following. Factors Product Sum Conclusion -1, No 1, No -, Yes, 6 -, -, We don t have to continue as we have found the correct combination.. Rewrite. x 8x + x x 6x + 5. Factor by grouping: x x 6x + (x x) + ( 6x + ) x (x ) (x ) (x )(x ) Answer: x 8x + (x )(x ) 1. x + x 1. x + 8x + 5. x + 5x +. x 7x 5. 5x 8x + 6. x 11x + 6 Special Factoring: The Difference of Two Squares y ( x y) ( x y ) or x y ( x + y ) ( x y ) x + Examples Solutions c) Factor: x 5 First, recognize that the quadratic is a difference of two squares. x 5 x 5 (x 5) (x + 5) Check: (x 5) (x + 5) x 5x + 5x 5 x 5 d) Factor: x 9 First, recognize that the quadratic is a difference of two squares. x 9 (x) 7 (x 7) (x + 7) Check: (x 7) (x + 7) x 1x + 1x 9 x 9 e) Factor: 81x 16y First, recognize that the quadratic is a difference of two squares. 81x 16y (9x ) (y ) (9x y ) (9x + y ) Notice that the first factor is also a difference of two squares. Thus, the quadratic can be factored further. 81x 16y (9x ) (y ) (9x y ) (9x + y ) (x y) (x + y) (9x + y ) Answer: 81x 16y (x y) (x + y) (9x + y )

3 10 CHAPTER Algebra Check Your Progress.6 Factor each of the following. 7. 5x t 100 Solving Quadratic Equations Case 1: Solving Quadratic Equations By Factoring The quadratic must be factorable. 1. Write the equation in standard form: ax + bx + c 0. Factor the quadratic.. Set each factor equal to zero (0).. Solve the resulting equations to find the solutions to the quadratic equation. Examples Solutions f) Solve: x 17x Notice that the quadratic equation is already in standard form. Factor the quadratic on the left side of the equation. x 17x (x )(x 5) 0 Set each factor equal to zero and solve. x 0 x 5 0 x x 5 Check by substituting the x values in the original equation. (5) 17(5) + 0 (5) g) Solve: 9x 5 0 Notice that the quadratic equation is already in standard form. Factor the quadratic on the left side of the equation. 9x 5 0 ( x ) - ( 5 ) 0 ( x 5 )(x + 5) 0 Set each factor equal to zero and solve. x 5 0 x x x

4 SECTION.6 Factoring and the Quadratic Equation 11 Example Solution h) Solve: x + 9x - First, write the quadratic equation in standard form by adding to both sides. Then, factor the quadratic on the left side of the equation. y + 9y - y + 9y + -+ y + 9y + 0 (y + 1)(y + ) 0 Set each factor equal to zero and solve. y y y y Check Your Progress.6 Find the solutions. 9. x x 10. x + 10x x 5x 0 1. x + 8x t x 5x Case : Solving Quadratic Equations Using the Quadratic Formula We can use the quadratic formula to find the roots to any quadratic equation. 1. Write the equation in standard form: ax + bx + c 0. Identify a, b, and c.. Use the quadratic formula:. Simplify. b x ± b a ac

5 1 CHAPTER Algebra Examples i) Find the real roots: 1x 5x 0 Solutions Notice that the quadratic equation is already in standard form. a 1, b 5, c. Substitute these values into the quadratic b ± formula and simplify. x b a ac x ( 5) ± ( 5) ( 1)( ) ( 1) 5 ± x 18 5 ± ± x j) Find the real roots: x + x 15 First, write the quadratic equation in standard form by subtracting 15 from both sides of the equation. x + x 15 0 a, b 1, c 15. Substitute these values into the quadratic formula b ± and simplify. x b a ac x 1 () 1 ± () 1 ( )( 15) ( ) ± x 5 1 ± ± x Check Your Progress.6 For Questions 15 0, find the roots to the equation. 15. x 10x x 6x x + x m + 6m 19. 7x + 6x x + 10x -1

6 SECTION.6 Factoring and the Quadratic Equation 1 See If You Remember SECTIONS Simplify: 5π + 9π. Simplify: Simplify: 9 7. ( ) (. 10 ) Simplify: Simplify: 8y y + 1y 7 7. Solve: ( t + 1) [ t + ( t ) ] 8. Name the property: x + (y + z) x + (z + y) 9. Solve: (y 1) + 5y < 7y 10. Find f (), if f (x) -x + x Find the solution set. x y 15 x y 1

7 1 CHAPTER Algebra 1. Graph: y -x + 1. Graph: x y 5 1. Graph the solution: -x + y >