Copyright © 2016, 2012, 2008 Pearson Education, Inc. 1 Factoring and Applications Chapter 5

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 2 Solving Quadratic Equations Using the Zero-Factor Property 1. Solve quadratic equations using the zero-factor property. 2. Solve other equations using the zero-factor property. 5.5

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 3 Quadratic Equation A quadratic equation (in x here) can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers and a 0. The given form is called standard form. Examples: x 2 + 5x + 6 = 0, 2x 2 – 5x = 3, x 2 = 4

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 4 Objective 1 Solve quadratic equations using the zero-factor property.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 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 be 0, but both may be 0.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 6 Classroom Example 1 or Using the Zero-Factor Property

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 7 Solve each equation. b. x(2x + 4) = 0 Check by substituting these values into the original equation. The solution set is {0, –2}. Classroom Example 1 Using the Zero-Factor Property (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 8 Solve each equation. a. x 2 + 2x = 8 Write the equation in standard form x 2 + 2x – 8 = 0 Factor the equation. Find two numbers whose product is –8 and whose sum is 2. Both values check, so the solution set is {–4, 2}. Classroom Example 2 Solving Quadratic Equations

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 9 Solve each equation. b. x 2 = x + 30 Write the equation in standard form. x 2 – x – 30 = 0 Factor the equation. Find two numbers whose product is –30 and whose sum is –1. Both values check, so the solution set is {–5, 6}. Classroom Example 2 Solving Quadratic Equations (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 10 Solving a Quadratic Equation Using the Zero-Factor Property Step 1 Write the equation in standard form—that is, with all terms on one side of the equality symbol in descending powers of the variable and 0 on the other side. Step 2 Factor completely. Step 3 Apply the zero-factor property. Set each factor with a variable equal to 0. Step 4 Solve the resulting equations. Step 5 Check each result in the original equation. Write the solution set.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 11 Solve 3m 2 – 9m = 30. Check each result to verify that the solution set is {–2, 5}. Classroom Example 3 Solving a Quadratic Equation (Common Factor)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 12 Solve each equation. a. 49x 2 – 9 = 0 The solution set is Classroom Example 4 or Solving Quadratic Equations

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 13 Solve each equation. b. x 2 = 3x The solution set is {0, 3}. Classroom Example 4 Solving Quadratic Equations (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 14 Solve each equation. c. x(4x + 7) = 2 Classroom Example 4 The solution set is Solving Quadratic Equations (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 15 Solve. Because the two factors are identical, they both lead to the same solution, called a double solution. Classroom Example 5 The solution set is {–8}. Solving Quadratic Equations (Double Solutions)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 16 Objective 2 Solve other equations using the zero-factor property.

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 17 Solve other equations using the zero- factor property. We can also use the zero-factor property to solve equations that involve more than two factors with variables. (These equations will have at least one term greater than second degree. They are not quadratic equations.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 18 Solve each equation. a. 2x 3 – 50x = 0 Check by substituting each solution into the original equation. The solution set is {–5, 0, 5}. Classroom Example 6 Solving Equations with More Than Two Variable Factors

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 19 Solve each equation. b. Check to verify that the solution set is Classroom Example 6 Solving Equations with More Than Two Variable Factors (cont.)

Copyright © 2016, 2012, 2008 Pearson Education, Inc. 20 Solve. Check to verify that the solution set is {0, 5}. Classroom Example 7 Solving an Equation Requiring Multiplication before Factoring