d 2 8d 16: The constant, 16, is obtained by 1) fi nding half of the coeffi cient of d; then 2) squaring the result. 1 2 (8) 4 (4)2 16 (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 x 2 bx to get x2 bx a1 2 bb 2 . The factored form is ax 1 2 bb 2 . The coefficient of the squared term must be 1 before you complete the square! sq EXAMPLE 5 Complete the square for each expression to obtain a perfect square trinomial. Then, factor. a) y2 6y b) t2 14t Solution a) Find the constant needed to complete the square for y2 6y. Step 1: Find half of the coeffi cient of y: 1 2 (6) 3 Step 2: Square the result: 32 9 Step 3: Add 9 to y2 6y: y2 6y 9 The perfect square trinomial is y2 6y 9. The factored form is (y 3)2. b) Find the constant needed to complete the square for t2 14t. Step 1: Find half of the coeffi cient of t: 1 2 (14) 7 Step 2: Square the result: (7)2 49 Step 3: Add 49 to t 2 14t: t 2 14t 49 The perfect square trinomial is t 2 14t 49. The factored form is (t 7)2. www.mhhe.com/messersmith SECTION 10.1 The Square Root Property and Completing the Square 617
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