1. Chapter 6 Section 5

2. Objectives 1 Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Quadratic Equations by Factoring Solve quadratic equations by factoring. Solve other equations by factoring. 6.5 2

3. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving Quadratic Equations by Factoring. Quadratic Equation A quadratic equation is an equation that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers, with a ≠ 0. The form ax 2 + bx + c = 0 is the standard form of a quadratic equation. For example, and are all quadratic equations, but only x 2 + 5x +6 = 0 is in standard form. Until now, we have factored expressions, including many quadratic expressions. In this section we see how we can use factored quadratic expressions to solve quadratic equations. Slide 6.5-3

4. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 1 Solve quadratic equations by factoring. Slide 6.5-4

5. Copyright © 2012, 2008, 2004 Pearson Education, Inc. We use the zero-factor property to solve a quadratic equation by factoring. Solve quadratic equations by factoring. Slide 6.5-5 Zero-Factor Property If a and b are real numbers and if ab = 0, then a = 0 or b = 0. That is, if the product of two numbers is 0, then at least one of the numbers must be 0. One number must, but both may be 0.

6. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve. Solution : or Slide 6.5-6 EXAMPLE 1 Using the Zero-Factor Property

7. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve. Solution: or Slide 6.5-7 EXAMPLE 2 Solving Quadratic Equations

8. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solving a Quadratic Equation by Factoring Step 1: Write the equation in standard form — that is, with all terms on one side of the equals sign in descending power of the variable and 0 on the other side. Step 2: Factor completely. Step 3: Use the zero-factor property to set each factor with variable equal to 0, and solve the resulting equations. Step 4: Check each solution in the original equation. Slide 6.5-8 Solve quadratic equations by factoring. (cont’d)

9. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Solve 3m 2 − 9m = 30. A common error is to include the common factor 3 as a solution. Only factors containing variables lead to solutions. Slide 6.5-9 EXAMPLE 3 Solving a Quadratic Equation with a Common Factor

10. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Solve. Slide 6.5-10 EXAMPLE 4 Solving Quadratic Equations

11. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Solve. Slide 6.5-11 EXAMPLE 4 Solving Quadratic Equations (cont’d)

12. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solution: Solve. Slide 6.5-12 EXAMPLE 5 Solving Quadratic Equations with Double Solutions There is no need to write the same number more than once in a solution set when a double solution occurs.

13. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Objective 2 Solve other equations by factoring. Slide 6.5-13

14. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve. Solution: Slide 6.5-14 EXAMPLE 6 Solving Equations with More than Two Variable Factors

15. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve. Solution: Slide 6.5-15 EXAMPLE 6 Solving Equations with More Than Two Variable Factors (cont’d)

16. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Solve. Solution: Slide 6.5-16 EXAMPLE 7 Solving an Equation Requiring Multiplication before Factoring