Quadratics and inequalities

X 2 4 x 4 12 4 one half of 4 is 2 and 2 2 4 x 2 2 8 x

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x 2 4 x 4 12 4 One-half of 4 is 2, and ( 2) 2 4. ( x 2) 2 8 x 2 8 Even-root property x 2 i 8 2 2 i 2 Check these values in the original equation. The solution set is 2 2 i 2 . Now do Exercises 67–76 Warm-Ups True or false? Explain your answer. 1. Completing the square means drawing the fourth side. 2. The equation ( x 3) 2 12 is equivalent to x 3 2 3. 3. Every quadratic equation can be solved by factoring. 4. The trinomial x 2 4 3 x 1 9 6 is a perfect square trinomial. 5. Every quadratic equation can be solved by completing the square. 6. To complete the square for 2 x 2 6 x 4, add 9 to each side. 7. (2 x 3)(3 x 5) 0 is equivalent to x 3 2 or x 5 3 . 8. In completing the square for x 2 3 x 4, add 9 4 to each side. 9. The equation x 2 8 is equivalent to x 2 2. 10. All quadratic equations have two distinct complex solutions. 10.1 Exercises Boost your GRADE at mathzone.com! Practice Problems Self-Tests Videos Net Tutor e-Professors Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences. 1. What are the three methods discussed in this section for solving a quadratic equation? Calculator Close-Up The answer key (ANS) can be used to check imaginary answers as shown here. dug22241_ch10a.qxd 11/10/2004 18:30 Page 626
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10-11 10.1 Factoring and Completing the Square 627 2. Which quadratic equations can be solved by the even-root property? 3. How do you find the last term for a perfect square trinomial when completing the square? 4. How do you complete the square when the leading coefficient is not 1? Solve by factoring. See Example 1. 5. x 2 x 6 0 6. x 2 6 x 8 0 7. a 2 2 a 15 8. w 2 2 w 15 9. 2 x 2 x 3 0 10. 6 x 2 x 15 0 11. y 2 14 y 49 0 12. a 2 6 a 9 0 13. a 2 16 0 14. 4 w 2 25 0 Use the even-root property to solve each equation. See Example 2. 15. x 2 81 16. x 2 9 4 17. x 2 1 9 6 18. a 2 32 19. ( x 3) 2 16 20. ( x 5) 2 4 21. ( z 1) 2 5 22. ( a 2) 2 8 23. w 3 2 2 7 4 24. w 2 3 2 5 9 Find the perfect square trinomial whose first two terms are given. See Example 3. 25. x 2 2 x 26. m 2 14 m 27. x 2 3 x 28. w 2 5 w 29. y 2 1 4 y 30. z 2 3 2 z 31. x 2 2 3 x 32. p 2 6 5 p Factor each perfect square trinomial. See Example 4. 33. x 2 8 x 16 34. x 2 10 x 25 35. y 2 5 y 2 4 5 36. w 2 w 1 4 37. z 2 4 7 z 4 4 9 38. m 2 6 5 m 2 9 5 39. t 2 3 5 t 1 9 00 40. h 2 3 2 h 1 9 6 Solve by completing the square. See Examples 5–7. Use your calculator to check. 41. x 2 2 x 15 0 42. x 2 6 x 7 0 43. 2 x 2 4 x 70 44. 3 x 2 6 x 24 45. w 2 w 20 0 46. y 2 3 y 10 0 47. q 2 5 q 14 48. z 2 z 2 49. 2 h 2 h 3 0 50. 2 m 2 m 15 0 51. x 2 4 x 6 52. x 2 6 x 8 0 53. x 2 8 x 4 0 54. x 2 10 x 3 0 55. 4 x 2 4 x 1 0 dug22241_ch10a.qxd 11/10/2004 18:30 Page 627
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628 Chapter 10 Quadratic Equations and Inequalities 10-12 56. 4 x 2 4 x 2 0 57. 2 x 2 3 x 4 0 58. 2 x 2 5 x 1 0 Solve each equation by an appropriate method. See Examples 8 and 9. 59. 2 x 1 x 1 60. 2 x 4 x 14 61. w w 2 1 62. y 1 y 2 1 63. t t 2 2 t t 3 64. z z 3 5 z 3 z 1 65. x 2 2 4 x 1 0 66. x 1 2 3 x 1 0 Use completing the square to find the imaginary solutions to each equation. See Example 10.
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