382 CHAPTER 6 Algebra, Graphs, and Functions SOLVING QUADRATIC EQUATIONS BY FACTORING 1. Use the addition or subtraction property so that one side of the equation is equal to 0. 2. Factor the side of the equation not equal to 0. 3. Use the zero-factor property to solve the equation. PROCEDURE Examples 8 and 9 illustrate this procedure. Example 8 Solving a Quadratic Equation by Factoring Solve the equation x x8 15. 2 − = − Solution First add 15 to both sides of the equation so the right-hand side of the equation is equal to 0. − = − − + = − + − + = x x x x x x 8 15 8 15 15 15 8 15 0 2 2 2 Factor the left-hand side of the equation. The goal is to determine two numbers whose product is 15 and whose sum is 8. − Since the product of the numbers is positive and the sum of the numbers is negative, the two numbers must both be negative. The numbers are 3− and 5. − Note that ( 3)( 5) 15 − − = and 3 (5) 8. − + − = − − + = − − = x x x x 8 15 0 ( 3)( 5) 0 2 Now use the zero-factor property to determine the solution. Set each factor equal to 0 and solve. − = = − = = x x x x 3 0 3 or 5 0 5 The solutions are 3 and 5. 7 Now try Exercise 45 Timely Tip Recall that every factoring problem can be checked by multiplying the factors. The product of the factors should be identical to the original expression that was factored. If we wished to check the factoring of Example 9, we would multiply x x (3 2)( 3). + − Since the product of the factors is x x 3 7 6, 2 − − which is the expression we started with, our factoring is correct. Example 9 Solving a Quadratic Equation by Factoring Solve the equation x x 3 7 6 0. 2 − − = Solution x x 3 7 6 2 − − factors into x x (3 2)( 3). + − Thus, we write − − = + − = x x x x 3 7 6 0 (3 2)( 3) 0 2 + = − = = − = = − x x x x x 3 2 0 or 3 0 3 2 3 2 3 The solutions are 2 3 − and 3. 7 Now try Exercise 51 Solving Quadratic Equations by Using the Quadratic Formula Not all quadratic equations can be solved by factoring. The quadratic formula is a formula that can be used to solve any quadratic equation. When a quadratic equation cannot be easily solved by factoring, we can use the quadratic formula to solve the equation.
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