133 1.4 Quadratic Equations (c) 1x - 422 = 12 x - 4 = {212 Generalized square root property x = 4{212 Add 4. x = 4{223 212 = 24 # 3 = 223 CHECK 1x - 422 = 12 Original equation EXAMPLE 2 Using the Square Root Property Solve each quadratic equation. (a) x2 = 17 (b) x2 = -25 (c) 1x - 422 = 12 SOLUTION (a) x2 = 17 x = {217 Square root property The solution set is E {217 F. (b) x2 = -25 x = {2-25 Square root property x = {5i 2-1 = i The solution set is 5{5i6. A4 + 223 - 4B 2 ≟12 Let x = 4 + 223. A223 B 2 ≟12 22 # A 23 B 2 ≟12 12 = 12 ✓ True A4 - 223 - 4B 2 ≟12 Let x = 4 - 223. A -223 B 2 ≟12 1-222 # A 23 B 2 ≟12 12 = 12 ✓ True The solution set is E4{223F . S Now Try Exercises 27, 29, and 31. Completing the Square Any quadratic equation can be solved by the method of completing the square, summarized in the box below. While this method may seem tedious, it has several useful applications, including analyzing the graph of a parabola and developing a general formula for solving quadratic equations. Solving a Quadratic Equation Using Completing the Square To solve ax2 + bx + c = 0, where a≠0, using completing the square, follow these steps. Step 1 If a≠1, divide each side of the equation by a. Step 2 Rewrite the equation so that the constant term is alone on one side of the equality symbol. Step 3 Square half the coefficient of x, and add this square to each side of the equation. Step 4 Factor the resulting trinomial as a perfect square, and combine like terms on the other side. Step 5 Use the square root property to complete the solution.
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