Quadratics and inequalities

10 20 8 6 using the formula the quadratic formula

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10 20 8 6 Using the Formula The quadratic formula solves any quadratic equation. Simply identify a , b , and c and insert those numbers into the formula. Note that if b is positive then b (the opposite of b ) is a negative number. If b is negative, then b is a positive number. dug22241_ch10a.qxd 11/10/2004 18:30 Page 631
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Because the solutions to the equations in Examples 1 and 2 were rational num- bers, these equations could have been solved by factoring. In Example 3 the solutions are irrational. 632 Chapter 10 Quadratic Equations and Inequalities 10-16 In this form we get a 4, b 12, and c 9. x Because b 12, b 12. 12 1 8 44 144 12 8 0 1 8 2 3 2 Check 3 2 in the original equation. The solution set is 3 2 . Now do Exercises 13–18 12 ( 12) 2 4(4)(9) 2(4) Calculator Close-Up Note that the single solution to 4 x 2 12 x 9 0 corresponds to the single x -intercept for the graph of y 4 x 2 12 x 9. 10 2 2 4 E X A M P L E 3 Two irrational solutions Solve 2 x 2 6 x 3 0. Solution Let a 2, b 6, and c 3 in the quadratic formula: x 6 4 36 24 6 4 12 6 4 2 3 2 ( 3 2 2 3 ) 3 2 3 Check these values in the original equation. The solution set is 3 2 3 . Now do Exercises 19–24 6 (6) 2 4(2)(3) 2(2) Calculator Close-Up The two irrational solutions to 2 x 2 6 x 3 0 correspond to the two x -intercepts for the graph of y 2 x 2 6 x 3. 5 3 5 5 dug22241_ch10a.qxd 11/10/2004 18:30 Page 632
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You have learned to solve quadratic equations by four different methods: the even-root property, factoring, completing the square, and the quadratic formula. The even-root property and factoring are limited to certain special equations, but you should use those methods when possible. Any quadratic equation can be solved by completing the square or using the quadratic formula. Because the quadratic formula is usually faster, it is used more often than completing the square. However, complet- ing the square is an important skill to learn. It will be used in the study of conic sec- tions later in this text. 10-17 10.2 The Quadratic Formula 633 E X A M P L E 4 Two imaginary solutions, no real solutions Find the complex solutions to x 2 x 5 0. Solution Let a 1, b 1, and c 5 in the quadratic formula: x 1 2 19 1 2 i 19 Check these values in the original equation. The solution set is 1 2 i 19 . There are no real solutions to the equation. Now do Exercises 25–30 1 (1) 2 4(1)(5) 2(1) Calculator Close-Up Because x 2 x 5 0 has no real solutions, the graph of y x 2 x 5 has no x -intercepts. 10 2 6 6 Methods for Solving ax 2 bx c 0 Method Comments Examples Even-root Use when b 0. ( x 2) 2 8 property x 2 8 Factoring Use when the polynomial x 2 5 x 6 0 can be factored. ( x 2)( x 3) 0 Quadratic Solves any quadratic x 2 5 x 3 0 formula equation x Completing Solves any quadratic equation, x 2 6 x 7 0 the square but quadratic formula is faster x 2 6 x 9 7 9 ( x 3) 2 2 5 25 4(3) 2 Number of Solutions The quadratic equations in Examples 1 and 3 had two real solutions each. In each of those examples the value of b 2 4 ac was positive. In Example 2 the quadratic equation had only one solution because the value of b 2 4 ac was zero. In Example 4 dug22241_ch10a.qxd 11/10/2004 18:30 Page 633
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