1. 6.5 – Solving Equations with Quadratic Techniques

2. Quadratic equations are in the form: ax 2 + bx + c,

3. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers

4. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs.

5. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2

6. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13

7. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9

8. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9

9. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9 2x 2 + 8x

10. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9 2x 2 + 8x 2x 2 + 8x + 0

11. Quadratic equations are in the form: ax 2 + bx + c, where a, b, & c are integers exs. x 2 + 5x + 2 2x 2 – 18x + 13 x 2 – 9 x 2 + 0x – 9 2x 2 + 8x 2x 2 + 8x + 0 NOTE: Must have the “x 2 ” term to be a quadratic!

12. Ex. 1 Write each expression in quadratic form, if possible.

13. a. x 4 + 13x 2 + 36

14. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2

15. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 )

16. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36

17. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36

18. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625

19. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2

20. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625

21. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16

22. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2

23. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2 – 9(x ¼ )

24. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2 – 9(x ¼ ) + 16

25. Ex. 1 Write each expression in quadratic form, if possible. a. x 4 + 13x 2 + 36 (x 2 ) 2 + 13(x 2 ) + 36 b. 16x 6 – 625 (4x 3 ) 2 – 625 c. x ½ – 9x ¼ + 16 (x ¼ ) 2 – 9(x ¼ ) + 16

26. Ex. 2 Solve each equation. a. x 4 = 16

27. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16

28. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0

29. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0

30. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0

31. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0

32. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 )(x 2 ) = 0

33. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0

34. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0

35. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0

36. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0

37. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x )(x ) = 0

38. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0

39. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0

40. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2

41. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

42. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

43. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

44. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 +2 +2 -2 -2 x = 2 OR x = -2

45. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 x = 2 OR x = -2

46. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 √x 2 = √-4 x = 2 OR x = -2

47. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 √x 2 = √-4 x = 2 OR x = -2 OR x = ±2i

48. Ex. 2 Solve each equation. a. x 4 = 16 -16 -16 x 4 – 16 = 0 (x 2 ) 2 – 16 = 0 ()() = 0 (x 2 – 4)(x 2 + 4) = 0 x 2 – 4 = 0 OR x 2 + 4 = 0 ( )( ) = 0 OR ( )( ) = 0 (x – 2)(x + 2) = 0 OR x 2 + 4 = 0 x – 2 = 0 OR x + 2 = 0 OR x 2 = -4 +2 +2 -2 -2 √x 2 = √-4 x = 2 OR x = -2 OR x = ±2i

49. b. x 4 + 11x 2 + 18 = 0

51. (x 2 ) 2 + 11(x 2 ) + 18 = 0

52. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0

53. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 )(x 2 ) = 0

54. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0

55. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0

56. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0

57. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0

58. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0

59. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 (x + )(x + ) = 0

60. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 x 2 + 9 = 0

61. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9

62. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9

63. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9

64. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

65. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

66. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 (x +)(x + ) = 0 x 2 + 9 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

67. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 x 2 = -9 √x 2 = √-9 x = ±3i

68. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 √x 2 = √-9 x = ±3i

69. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 x = ±3i

70. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 OR √x 2 = √-2 x = ±3i

71. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 OR √x 2 = √-2 x = ±3i OR x = ±i√2

72. b. x 4 + 11x 2 + 18 = 0 (x 2 ) 2 + 11(x 2 ) + 18 = 0 ()() = 0 (x 2 + 9)(x 2 + 2) = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 ( )( ) = 0 ()() = 0 x 2 + 9 = 0 OR x 2 + 2 = 0 -9 -9 -2 -2 x 2 = -9 OR x 2 = -2 √x 2 = √-9 OR √x 2 = √-2 x = ±3i OR x = ±i√2

73. c. x ½ – 6x ¼ – 16 = 0

75. (x ¼ ) 2 – 6(x ¼ ) – 16 = 0

76. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0

77. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ )(x ¼ ) = 0

78. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0

79. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0

80. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0

81. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0

82. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0

83. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8

84. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8

85. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4

86. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

87. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

88. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

89. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

90. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 (x ¼ ) 4 = (8) 4 x = 4096

91. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 x = 4096

92. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096

93. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096 OR x = 16

94. c. x ½ – 6x ¼ – 16 = 0 (x ¼ ) 2 – 6(x ¼ ) – 16 = 0 ()() = 0 (x ¼ – 8)(x ¼ + 2) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 ( )( ) = 0 ()( ) = 0 x ¼ – 8 = 0 OR x ¼ + 2 = 0 +8 +8 -2 -2 x ¼ = 8 OR x ¼ = -2 (x ¼ ) 4 = (8) 4 OR (x ¼ ) 4 = (-2) 4 x = 4096 OR x = 16