6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 381 Note that in Example 6 we first tried factors of the form x x (2 )(3 ). If we had not determined the correct factors using them, we would have tried x x (6 )( ). Solving Quadratic Equations by Factoring In Section 6.1, we solved linear, or first-degree, equations. In those equations, the exponent on all variables was 1. We now solve quadratic equations. A quadratic equation is one in which the greatest exponent on a variable is 2. The standard form of a quadratic equation in one variable is shown in the box. Standard Form of a Quadratic Equation + + = ≠ ax bx c a 0, 0 2 Note that in the standard form of a quadratic equation, the greatest exponent on x is 2 and the right side of the equation is equal to zero. In this section, we will solve quadratic equations by factoring and by the quadratic formula. To solve quadratic equations by factoring, we will use the zero-factor property. Zero-Factor Property a b a b If 0, then 0 or 0. ⋅ = = = The zero-factor property indicates that if the product of two factors is 0, then at least one of the factors must have a value of 0. Example 7 Using the Zero-Factor Property Solve the equation x x ( 4)( 3) 0. + − = Solution When we use the zero-factor property, we set each individual factor equal to 0 and solve each resulting equation for x. x x ( 4)( 3) 0 + − = x x x x 4 0 or 3 0 4 3 + = − = = − = Thus, the solutions are 4− and 3. CHECK: x 4 = − x 3 = x x ( 4)( 3) 0 ( 4 4)( 4 3) 0 0( 7) 0 0 0 + − = − + − − = − = = True x x ( 4)( 3) 0 (3 4)(3 3) 0 7(0) 0 0 0 + − = + − = = = True 7 Now try Exercise 37
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