Quadratics and inequalities

A a single rational solution b two rational solutions

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a) a single rational solution b) two rational solutions c) two irrational solutions d) two imaginary solutions 108. Exploration In the next section we will solve ax 2 bx c 0 for x by completing the square. Try it now without looking ahead. Graphing Calculator Exercises For each equation, find approximate solutions rounded to two decimal places. 109. x 2 7.3 x 12.5 0 110. 1.2 x 2 x 2 0 111. 2 x 3 20 x 112. x 2 1.3 x 22.3 x 2 Figure for Exercise 104 0 0 5 1 2 10 15 20 25 Time (sec) Height (ft) In this Section Developing the Formula Using the Formula Number of Solutions Applications 10.2 The Quadratic Formula Completing the square from Section 10.1 can be used to solve any quadratic equation. Here we apply this method to the general quadratic equation to get a formula for the solutions to any quadratic equation. Developing the Formula Start with the general form of the quadratic equation, ax 2 bx c 0. dug22241_ch10a.qxd 11/10/2004 18:30 Page 629
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Assume a is positive for now, and divide each side by a : ax 2 a bx c 0 a x 2 b a x a c 0 x 2 b a x a c Subtract a c from each side. One-half of b a is 2 b a , and 2 b a squared is 4 b a 2 2 : x 2 b a x 4 b a 2 2 a c 4 b a 2 2 Factor the left-hand side and get a common denominator for the right-hand side: x 2 b a 2 4 b a 2 2 4 4 a a c 2 a c ( ( 4 4 a a ) ) 4 4 a a c 2 x 2 b a 2 b 2 4 a 2 4 ac x 2 b a b 2 4 a 2 4 ac Even-root property x 2 a b Because a 0, 4 a 2 2 a . x We assumed a was positive so that 4 a 2 2 a would be correct. If a is negative, then 4 a 2 2 a , and we get x 2 a b . However, the negative sign can be omitted in 2 a because of the symbol preced- ing it. For example, the results of 5 ( 3) and 5 3 are the same. So when a is negative, we get the same formula as when a is positive. It is called the quadratic formula. The Quadratic Formula The solution to ax 2 bx c 0, with a 0, is given by the formula x b 2 a b 2 4 ac . b 2 4 ac 2 a b b 2 4 ac 2 a b 2 4 ac 2 a 630 Chapter 10 Quadratic Equations and Inequalities 10-14 dug22241_ch10a.qxd 11/10/2004 18:30 Page 630
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10-15 10.2 The Quadratic Formula 631 E X A M P L E 1 Two rational solutions Solve x 2 2 x 15 0 using the quadratic formula. Solution To use the formula, we first identify the values of a , b , and c : 1 x 2 2 x 15 0 a b c The coefficient of x 2 is 1, so a 1. The coefficient of 2 x is 2, so b 2. The constant term is 15, so c 15. Substitute these values into the quadratic formula: x 2 2 4 60 2 2 64 2 2 8 x 2 2 8 3 or x 2 2 8 5 Check 3 and 5 in the original equation. The solution set is 5, 3 . Now do Exercises 7–12 To identify a , b , and c for the quadratic formula, the equation must be in the standard form ax 2 bx c 0. If it is not in that form, then you must first rewrite the equation. CAUTION 2 2 2 4(1)( 15) 2(1) E X A M P L E 2 One rational solution Solve 4 x 2 12 x 9 by using the quadratic formula. Solution Rewrite the equation in the form ax 2 bx c 0 before identifying a , b , and c : 4 x 2 12 x 9 0 Calculator Close-Up Note that the two solutions to x 2 2 x 15 0 correspond to the two x -intercepts for the graph of y x 2 2 x 15.
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