Read the explanations, follow the examples, take notes, and complete the You Trys. The next method we will learn for solving a quadratic equation is completing the square. But fi rst we need to review an idea presented in Section 7.4. A perfect square trinomial is a trinomial whose factored form is the square of a binomial. Some examples of perfect square trinomials are Perfect Square Trinomials Factored Form x2 6x 9 (x 3)2 d 2 14d 49 (d 7)2 In the trinomial x2 6x 9, x2 is called the quadratic term, 6x is called the linear term, and 9 is called the constant. 1 Complete the Square for an Expression of the Form x2 bx In a perfect square trinomial where the coeffi cient of the quadratic term is 1, the constant term is related to the coeffi cient of the linear term in the following way: If you fi nd half of the linear coeffi cient and square the result, you will get the constant term. x2 6x 9: The constant, 9, is obtained by 1) fi nding half of the coeffi cient of x 1 2 (6) 3 2) then squaring the result. 32 9 (the constant) d2 14d 49: The constant, 49, is obtained by 1) fi nding half of the coeffi cient of d 1 2 (14) 7 2) then squaring the result. (7)2 49 (the constant) We can generalize this procedure so that we can fi nd the constant needed to obtain the perfect square trinomial for any quadratic expression of the form x2 bx. Finding this perfect square trinomial is called completing the square because the trinomial will factor to the square of a binomial. Procedure Completing the Square for x2 bx To fi nd the constant needed to complete the square for x2 bx: Step 1: Find half of the coeffi cient of x: 1 2 b Step 2: Square the result: a1 2 bb 2 Step 3: Then add it to x2 bx to get x2 bx a1 2 bb 2 www.mhhe.com/messersmith SECTION 10.2 Solving Quadratic Equations by Completing the Square 613
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