Quadratics and inequalities

B one half of 5 is 5 2 and 5 2 squared is 2 4 5 so

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b) One-half of 5 is 5 2 , and 5 2 squared is 2 4 5 . So the perfect square trinomial is x 2 5 x 2 4 5 . c) Since 1 2 4 7 2 7 and 2 7 squared is 4 4 9 , the perfect square trinomial is x 2 4 7 x 4 4 9 . d) Since 1 2 3 2 3 4 and 3 4 2 1 9 6 , the perfect square trinomial is x 2 3 2 x 1 9 6 . Now do Exercises 25–32 Another essential step in completing the square is to write the perfect square trinomial as the square of a binomial. Recall that a 2 2 ab b 2 ( a b ) 2 and a 2 2 ab b 2 ( a b ) 2 . E X A M P L E 4 Factoring perfect square trinomials Factor each trinomial. a) x 2 12 x 36 b) y 2 7 y 4 4 9 c) z 2 4 3 z 4 9 dug22241_ch10a.qxd 11/10/2004 18:29 Page 620
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In Example 5 we use the skills that we practiced in Examples 2, 3, and 4 to solve the quadratic equation ax 2 bx c 0 with a 1 by the method of completing the square. 10-5 10.1 Factoring and Completing the Square 621 Solution a) The trinomial x 2 12 x 36 is of the form a 2 2 ab b 2 with a x and b 6. So x 2 12 x 36 ( x 6) 2 . Check by squaring x 6. b) The trinomial y 2 7 y 4 4 9 is of the form a 2 2 ab b 2 with a y and b 7 2 . So y 2 7 y 4 4 9 y 7 2 2 . Check by squaring y 7 2 . c) The trinomial z 2 4 3 z 4 9 is of the form a 2 2 ab b 2 with a z and b 2 3 . So z 2 4 3 z 4 9 z 2 3 2 . Now do Exercises 33–40 Helpful Hint To square a binomial use the follow- ing rule (not FOIL): Square the first term. Add twice the product of the terms. Add the square of the last term. E X A M P L E 5 Completing the square with a 1 Solve x 2 6 x 5 0 by completing the square. Solution The perfect square trinomial whose first two terms are x 2 6 x is x 2 6 x 9. So we move 5 to the right-hand side of the equation, then add 9 to each side to create a perfect square on the left side: x 2 6 x 5 Subtract 5 from each side. x 2 6 x 9 5 9 Add 9 to each side to get a perfect square trinomial. ( x 3) 2 4 Factor the left-hand side. x 3 4 Even-root property x 3 2 or x 3 2 x 1 or x 5 Check in the original equation: ( 1) 2 6( 1) 5 0 and ( 5) 2 6( 5) 5 0 The solution set is 1, 5 . Now do Exercises 41–48 Calculator Close-Up The solutions to x 2 6 x 5 0 correspond to the x -intercepts for the graph of y x 2 6 x 5. So we can check our solutions by graphing and using the TRACE fea- ture as shown here. 6 8 2 6 dug22241_ch10a.qxd 11/10/2004 18:29 Page 621
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All of the perfect square trinomials that we have used so far had a leading coefficient of 1. If a 1, then we must divide each side of the equation by a to get an equation with a leading coefficient of 1. The strategy for solving a quadratic equation by completing the square is stated in the following box. CAUTION 622 Chapter 10 Quadratic Equations and Inequalities 10-6 E X A M P L E 6 Strategy for Solving Quadratic Equations by Completing the Square 1. If a 1, then divide each side of the equation by a . 2. Get only the x 2 and the x terms on the left-hand side.
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