A50 APPENDIX Review 5 Solve Quadratic Equations by Completing the Square We now introduce the method of completing the square. The idea behind this method is to “adjust” the left side of a quadratic equation, + + = ax bx c 0, 2 so that it becomes a perfect square – that is, the square of a first-degree polynomial. For example, + + x x6 9 2 and − + x x4 4 2 are perfect squares because ( ) ( ) + + = + − + = − x x x x x x 6 9 3 and 4 4 2 2 2 2 2 How do we “adjust” the left side? We do it by adding the appropriate number to create a perfect square. For example, to make + x x6 2 a perfect square, add 9. Let’s look at several examples of completing the square when the coefficient of x2 is 1. Need to Review? Completing the square is discussed in Section A.3, p. A29. Start Add Result + x x4 2 4 ( ) + + = + x x x 4 4 2 2 2 + x x 12 2 36 ( ) + + = + x x x 12 36 6 2 2 − x x6 2 9 ( ) − + = − x x x 6 9 3 2 2 + x x 2 1 4 ( ) + + = + x x x 1 4 1 2 2 2 Start Add Result + x mx 2 ( ) m 2 2 ( ) ( ) + + = + x mx m x m 2 2 2 2 2 Do you see the pattern? Provided that the coefficient of x2 is 1, complete the square by adding the square of one-half of the coefficient of x. Now Work problem 101 The next example illustrates how the procedure of completing the square can be used to solve a quadratic equation. Solving a Quadratic Equation by Completing the Square Solve by completing the square: + + = x x5 4 0 2 Solution EXAMPLE 7 Always begin by writing the equation with the constant on the right side. + + = + = − x x x x 5 4 0 5 4 2 2 Since the coefficient of x2 is 1, complete the square on the left side by adding ( ) ⋅ = 1 2 5 25 4 . 2 Of course, in an equation, whatever is added to the left side must also be added to the right side. So, add 25 4 to both sides. ( ) + + = − + + = + = ± x x x x 5 25 4 4 25 4 5 2 9 4 5 2 9 4 2 2 Add 25 4 to both sides. Factor; simplify. Use the Square Root Method.
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