SECTION A.7 Complex Numbers; Quadratic Equations in the Complex Number System A65 Determining the Character of the Solutions of a Quadratic Equation Without solving, determine the character of the solutions of each equation. (a) + + = x x 3 4 5 0 2 (b) + + = x x 2 4 1 0 2 (c) − + = x x 9 6 1 0 2 Solution EXAMPLE 12 (a) Here = = a b 3, 4, and = c 5, so − = − ⋅ ⋅ = − b ac 4 16 4 3 5 44. 2 The solutions are two complex numbers that are not real and are conjugates of each other. (b) Here = = a b 2, 4, and = c 1, so − = − = b ac 4 16 8 8. 2 The solutions are two unequal real numbers. (c) Here = = − a b 9, 6, and = c 1, so − = − ⋅ ⋅ = b ac 4 36 4 9 1 0. 2 The solution is a repeated real number—that is, a double root. Now Work problem 79 Skill Building In Problems 11–58, perform the indicated operation, and write each expression in the standard form +a bi. Verify your results using a graphing utility. 11. ( ) ( ) − + + i i 2 3 6 8 12. ( ) ( ) + + − + i i 4 5 8 2 13. ( ) ( ) − + − − i i 3 2 4 4 14. ( ) ( ) − − − − i i 3 4 3 4 15. ( ) ( ) − − + i i 2 5 8 6 16. ( ) ( ) − + − − i i 8 4 2 2 17. ( ) − i 3 2 6 18. ( ) − + i 4 2 8 19. ( ) − + i i 3 7 6 20. ( ) − + i i 3 3 4 21. ( )( ) − + i i 3 4 2 22. ( )( ) + − i i 5 3 2 23. ( )( ) − − − + i i 5 5 24. ( ) ( ) − + + i i 3 3 25. − i 10 3 4 26. − i 13 5 12 27. +i i 2 28. − − i i 2 2 29. − + i i 6 1 30. + − i i 2 3 1 31. + i 1 2 3 2 2 32. − i 3 2 1 2 2 33. ( ) +i 1 2 34. ( ) −i 1 2 35. i23 36. i14 37. i 20 − 38. −i 23 39. − i 5 6 40. +i 4 3 41. − i i 6 4 3 5 42. − + i i 4 2 1 3 2 43. ( ) +i 1 3 44. ( ) + i3 1 4 45. ( ) + i i 1 7 2 46. ( ) + i i 2 1 4 2 47. + − − i i i i 8 6 4 2 48. + + + i i i i 7 5 3 49. −4 50. −9 51. −25 52. −64 53. −12 54. −18 55. −200 56. −45 57. ( )( ) + − i i 3 4 4 3 58. ( )( ) + − i i 4 3 3 4 In Problems 59–78, solve each equation in the complex number system. Check your results using a graphing utility. 59. + = x 4 0 2 60. − = x 4 0 2 61. − = x 16 0 2 62. + = x 25 0 2 63. − + = x x6 13 0 2 64. + + = x x4 8 0 2 65. − + = x x6 10 0 2 66. − + = x x2 5 0 2 67. − + = x x 25 10 2 0 2 68. + + = x x 10 6 1 0 2 69. + = x x 5 1 2 2 70. + = x x 13 1 6 2 71. + + = x x 1 0 2 72. − + = x x 1 0 2 73. − = x 64 0 3 74. + = x 27 0 3 75. = x 16 4 76. = x 1 4 77. + + = x x 13 36 0 4 2 78. + − = x x3 4 0 4 2 A.7 Assess Your Understanding Concepts and Vocabulary 1. True or False The square of a complex number is sometimes negative. 2. ( )( ) + − = i i 2 2 . 3. True or False In the complex number system, a quadratic equation has four solutions. 4. In the complex number + i 5 2 , the number 5 is called the part; the number 2 is called the part; the number i is called the . 5. True or False The conjugate of + i 2 5 is − − i 2 5 . 6. True or False All real numbers are complex numbers. 7. True or False If − i 2 3 is a solution of a quadratic equation with real coefficients, then − + i 2 3 is also a solution. 8. Multiple Choice Which of the following is the principal square root of −4? (a) − i2 (b) i2 (c) −2 (d) 2 9. Multiple Choice Which operation involving complex numbers requires the use of a conjugate? (a) division (b) multiplication (c) subtraction (d) addition 10. Multiple Choice Powers of i repeat every power. (a) second (b) third (c) fourth (d) fifth 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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