5.5 Solving Quadratic Equations by Factoring

Solving Quadratic Equations by Factoring. A quadratic equation is an equation that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers, with a ≠ 0. The form ax2 + bx + c = 0 is the standard form of a quadratic equation. For example, and are all quadratic equations, but only x2 + 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 5.5-3

Solve quadratic equations by factoring. Objective 1 Solve quadratic equations by factoring. Slide 5.5-4

Solve quadratic equations by factoring. We use the zero-factor property to solve a quadratic equation by factoring. 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. Slide 5.5-5

Using the Zero-Factor Property CLASSROOM EXAMPLE 1 Using the Zero-Factor Property Solve. Solution: or or Slide 5.5-6

Solving Quadratic Equations CLASSROOM EXAMPLE 2 Solving Quadratic Equations Solve each equation. Solution: or or Slide 5.5-7

Solving a Quadratic Equation by Factoring Solve quadratic equations by factoring. (cont’d) 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 5.5-8

CLASSROOM EXAMPLE 3 Solve 3m2 − 9m = 30. Solution: Solving a Quadratic Equation with a Common Factor Solve 3m2 − 9m = 30. Solution: A common error is to include the common factor 3 as a solution. Only factors containing variables lead to solutions. Slide 5.5-9

Solving Quadratic Equations CLASSROOM EXAMPLE 4 Solving Quadratic Equations Solve each equation. Solution: Slide 5.5-10

Solving Quadratic Equations (cont’d) CLASSROOM EXAMPLE 4 Solving Quadratic Equations (cont’d) Solve the equation. Solution: Slide 5.5-11

Solving Quadratic Equations with Double Solutions CLASSROOM EXAMPLE 5 Solving Quadratic Equations with Double Solutions Solve. Solution: There is no need to write the same number more than once in a solution set when a double solution occurs. Slide 5.5-12

Solve other equations by factoring. Objective 2 Solve other equations by factoring. Slide 5.5-13

CLASSROOM EXAMPLE 6 Solve the equation. Solution: Solving Equations with More than Two Variable Factors Solve the equation. Solution: Slide 5.5-14

CLASSROOM EXAMPLE 6 Solve the equation. Solution: Solving Equations with More Than Two Variable Factors (cont’d) Solve the equation. Solution: Slide 5.5-15

CLASSROOM EXAMPLE 7 Solve. Solution: Solving an Equation Requiring Multiplication before Factoring Solve. Solution: Slide 5.5-16