Lesson 6.3: Solving Quadratic Equations by Factoring, pg. 301 Goals:  To solve quadratic equations by factoring.  To write a quadratic equation with.

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Solving Quadratic Equations

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Lesson 6.3: Solving Quadratic Equations by Factoring, pg. 301 Goals:  To solve quadratic equations by factoring.  To write a quadratic equation with given roots.

Zero Product Property For any real numbers a and b, if ab = 0, then either a = 0 or b = 0, or both a and b equal zero. Ex. (x + 5)(x – 7) = 0 x + 5 = 0 and x – 7 = 0 x = -5 and x = 7

Example 1: Solve each equation by factoring. (When solving a quadratic equation by factoring it Must be set equal to ZERO) 1.x² = -4x x²+4x = 0 X(x + 4) = 0 X=0 and x+4=0 X=0 and x = -4 (ans.) 2.3x² = 5x + 2 3x² - 5x – 2=0 (3x + 1)(x – 2) = 0 3x + 1 = 0 and x – 2 =0 3x = -1 and x = 2 x = -1/3 and x = 2 x = -1/3 and x = 2 (ans.)

3.x ² - 6x = -9 x ² - 6x + 9=0 set equal to zero (x – 3)(x – 3) = 0 factor x – 3 = 0 Zero Product Property (Double Root, write factor once) x = 3 Solve for x

Example 2: Write a quadratic equation given roots. (Use the roots/solutions to write a quadratic equation) 1.-2/3 and 6 x = -2/3 and x = 6 set each root equal to x X+ 2/3 = 0 and x – 6 = 0 set equation equal to zero (x + 2/3)(x – 6 ) = 0 rewrite using concept of Zero Product Property x² - 6x + 2/3x – 4 = 0 FOIL METHOD x² - 16/3x – 4 = 0 combine like terms 3(x² - 16/3x – 4 = 0 ) mult. by 3 so that coefficients are integers 3x² - 16x – 12 = 0 use dist. Prop. (final answer)

2.-4 and 7 x = -4 and x = 7 set each root equal to x X +4 = 0 x – 7 =0 set equation equal to zero (x +4)( x – 7) = 0 rewrite using concept of Zero Product Property x² - 7x + 4x – 28 = 0 FOIL METHOD x² - 3x – 28 = 0 combine like terms (final answer)