386 CHAPTER 6 Algebra, Graphs, and Functions 63. x x4 2 0 2 − + = 2 2 ± 64. x x2 2 0 2 − − = 1 3 ± 65. x x2 3 0 2 − + = No real solution 66. x x 2 4 5 0 2 − + = No real solution 67. x x 3 9 5 0 2 + + = 9 21 6 − ± 68. x x 2 5 2 0 2 − − = 5 41 4 ± 69. x x 4 5 3 0 2 − − = 5 73 8 ± 70. x x 3 7 5 0 2 − − = 7 109 6 ± 71. x x 3 9 5 2 = − 9 21 6 ± 72. x x 4 7 1 2 = − + 7 65 8 − ± 73. x x4 1 0 2 − + = 2 3 ± 74. x x6 7 0 2 − + = 3 2 ± 75. x x 3 8 1 0 2 − + = 4 13 3 ± 76. + + = x x 2 4 1 0 2 2 2 2 − ± 77. x x 4 6 5 2 + = − No real solution 78. x x 5 4 2 2 − = − No real solution Problem Solving 79. Drying Time The time, t, in minutes, needed for clothes hanging on a line outdoors to dry, at a specific temperature, depends on the humidity, h. The time can be approximated by the equation t h h 2 80 40, 2 = + + where h is the humidity expressed as a decimal number. Determine the length of time required for clothing to dry if there is 60% humidity. 88.72 min 80. Height of a Diver A diver jumps from a platform diving board that is 32 feet above water. The height, h, of the diver above water, in feet, t seconds after jumping from the platform, can be determined by the equation h t t 16 20 32. 2 = − + + Determine the diver’s height above water 2 seconds after jumping from the platform. 8 ft 81. Garden and Walkway CJ’s garden is surrounded by a uniform-width walkway. The garden and walkway together cover an area of 320 square feet. If the dimensions of the garden are 12 feet by 16 feet, determine the width of the walkway. 2 ft 82. Basketball Court The basketball court in the Kieler gym is surrounded by a seating area of uniform width. The court and the seating area have a combined area of 6300 ft .2 If the dimensions of the court are 95 feet by 50 feet, determine the width of the seating area. 5 ft 83. Height of a Ball A ball is projected upward from the ground. Its height h, above ground, in feet, can be determined by the equation h t t 16 128 2 = − + where t is the number of seconds after the ball is projected. After how many seconds will the ball be 256 ft above ground? 4 seconds 84. Height of a Rocket A model rocket is launched from a hill 80 feet above sea level. The launch site is next to the ocean, and the rocket will fall into the ocean. The rocket’s height, h, above sea level can be determined by the equation h t t 16 64 80 2 = − + + where t is the number of seconds after the rocket is launched. Determine the time it takes the rocket to strike the ocean. 5 seconds Challenge Problems/Group Activities 85. a) Explain why solving the equation x x ( 4)( 7) 6 − − = by setting each factor equal to 6 is not correct. The zero-factor property cannot be used. b) Determine the correct solution to x x ( 4)( 7) 6. − − = 11 33 2 ± 86. The radicand in the quadratic formula, b ac 4 , 2 − is called the discriminant. How many real number solutions will the quadratic equation have if the discriminant is (a) greater than 0, (b) equal to 0, or (c) less than zero? a) Two b) One c) None 87. Write an equation that has solutions 1− and 3. x x2 3 0, 2 − − = other answers are possible Research Activities 88. Girolamo Cardano Italian mathematician Girolamo Cardano (1501–1576) is recognized for his skill in solving equations. Write a paper about his life and his contributions to mathematics, in particular his contribution to solving equations. 89. Wang Xiaotong Chinese mathematician Wang Xiaotong (580–640) developed a method for solving polynomial equations. This method was described later by British mathematician William George Horner and became known as Horner’s method. Write a paper about this method. Describe how the method came to be used to evaluate higher degree polynomials using computers. Shutterstock
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