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the quadratic equation had no real solutions because b 2 4 ac was negative. Because b 2 4 ac determines the kind and number of solutions to a quadratic equation, it is called the discriminant. Number of Solutions to a Quadratic Equation The quadratic equation ax 2 bx c 0 with a 0 has two real solutions if b 2 4 ac 0, one real solution if b 2 4 ac 0, and no real solutions (two imaginary solutions) if b 2 4 ac 0. 634 Chapter 10 Quadratic Equations and Inequalities 10-18 E X A M P L E 5 Using the discriminant Use the discriminant to determine the number of real solutions to each quadratic equation. a) x 2 3 x 5 0 b) x 2 3 x 9 c) 4 x 2 12 x 9 0 Solution a) For x 2 3 x 5 0, use a 1, b 3, and c 5 in b 2 4 ac : b 2 4 ac ( 3) 2 4(1)( 5) 9 20 29 Because the discriminant is positive, there are two real solutions to this quadratic equation. b) Rewrite x 2 3 x 9 as x 2 3 x 9 0. Then use a 1, b 3, and c 9 in b 2 4 ac : b 2 4 ac ( 3) 2 4(1)(9) 9 36 27 Because the discriminant is negative, the equation has no real solutions. It has two imaginary solutions. c) For 4 x 2 12 x 9 0, use a 4, b 12, and c 9 in b 2 4 ac : b 2 4 ac ( 12) 2 4(4)(9) 144 144 0 Because the discriminant is zero, there is only one real solution to this quadratic equation. Now do Exercises 31–46 Applications With the quadratic formula we can easily solve problems whose solutions are irrational numbers. When the solutions are irrational numbers, we usually use a calculator to find rational approximations and to check. dug22241_ch10a.qxd 11/10/2004 18:30 Page 634

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10-19 10.2 The Quadratic Formula 635 E X A M P L E 6 Area of a tabletop The area of a rectangular tabletop is 6 square feet. If the width is 2 feet shorter than the length, then what are the dimensions? Solution Let x be the length and x 2 be the width, as shown in Fig. 10.1. Because the area is 6 square feet and A LW , we can write the equation x ( x 2) 6 or x 2 2 x 6 0. Because this equation cannot be factored, we use the quadratic formula with a 1, b 2, and c 6: x 2 2 28 2 2 2 7 1 7 Because 1 7 is a negative number, it cannot be the length of a tabletop. If x 1 7, then x 2 1 7 2 7 1. Checking the product of 7 1 and 7 1, we get ( 7 1 )( 7 1 ) 7 1 6. The exact length is 7 1 feet, and the width is 7 1 feet. Using a calcula- tor, we find that the approximate length is 3.65 feet and the approximate width is 1.65 feet. Now do Exercises 75–94 2 ( 2) 2 4(1)( 6) 2(1) Figure 10.1 x 2 ft x ft Warm-Ups True or false? Explain your answer. 1. Completing the square is used to develop the quadratic formula. 2. For the equation 3 x 2 4 x 7, we have a 3, b 4, and c 7. 3. If dx 2 ex f 0 and d 0, then x . 4. The quadratic formula will not work on the equation x 2 3 0. 5. If a 2, b 3, and c 4, then b 2 4 ac 41. 6. If the discriminant is zero, then there are no imaginary solutions. 7. If b 2 4 ac 0, then ax 2 bx c 0 has two real solutions.
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