134 CHAPTER 1 Equations and Inequalities EXAMPLE 3 Using Completing the Square 1a =12 Solve x2 - 4x - 14 = 0. SOLUTION x2 - 4x - 14 = 0 Step 1 This step is not necessary because a = 1. Step 2 x2 - 4x = 14 Add 14 to each side. Step 3 x2 - 4x + 4 = 14 + 4 C 1 2 1-42D 2 = 4; Add 4 to each side. Step 4 1x - 222 = 18 Factor. Combine like terms. Step 5 x - 2 = {218 Square root property x = 2{218 Add 2 to each side. x = 2{322 218 = 29 # 2 = 322 The solution set is E2{322 F. S Now Try Exercise 41. Take both roots. EXAMPLE 4 Using Completing the Square 1a 312 Solve 9x2 - 12x + 9 = 0. SOLUTION 9x2 - 12x + 9 = 0 Step 1 x2 - 4 3 x + 1 = 0 Divide by 9 so that a = 1. Step 2 x2 - 4 3 x = -1 Subtract 1 from each side. Step 3 x2 - 4 3 x + 4 9 = -1 + 4 9 C 1 2 A - 4 3B D 2 = 4 9 ; Add 4 9 to each side. Step 4 ax - 2 3b 2 = - 5 9 Factor. Combine like terms. Step 5 x - 2 3 = {B- 5 9 Square root property x - 2 3 = { 25 3 i 3-5 9 = 2-5 29 = i 25 3 , or 25 3 i x = 2 3 { 25 3 i Add 2 3 to each side. The solution set is E 2 3 { 25 3 i F. S Now Try Exercise 47. The Quadratic Formula If we start with the equation ax2 + bx + c = 0, for a 70, and complete the square to solve for x in terms of the constants a, b, and c, the result is a general formula for solving any quadratic equation. ax2 + bx + c = 0 x2 + b a x + c a = 0 Divide each side by a. (Step 1) x2 + b a x = - c a Subtract c a from each side. (Step 2)
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