Quadratics and inequalities

67 x 2 2 x 5 68 x 2 4 x 5 69 x 2 6 x 11 70 x 2 8 x 19

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67. x 2 2 x 5 0 68. x 2 4 x 5 0 69. x 2 6 x 11 0 70. x 2 8 x 19 0 71. x 2 1 2 72. x 2 1 8 73. x 2 12 0 74. 3 x 2 21 0 75. 5 z 2 4 z 1 0 76. 2 w 2 3 w 2 0 Find all real or imaginary solutions to each equation. Use the method of your choice. 77. x 2 121 78. w 2 225 79. 4 x 2 25 0 80. 5 w 2 3 0 81. p 1 2 2 9 4 82. y 2 3 2 4 9 83. 5 t 2 4 t 3 0 84. 3 v 2 4 v 1 0 85. m 2 2 m 24 0 86. q 2 6 q 7 0 87. ( x 2) 2 9 88. (2 x 1) 2 4 89. x 2 x 6 0 90. x 2 x 12 0 91. x 2 6 x 10 0 92. x 2 8 x 17 0 93. 2 x 5 7 x 7 94. 7 x 29 x 3 95. 1 x x 1 1 1 4 96. 1 x 1 2 x 1 2 If the solution to an equation is imaginary or irrational, it takes a bit more effort to check. Replace x by each given number to verify each statement. 97. Both 2 3 and 2 3 satisfy x 2 4 x 1 0. 98. Both 1 2 and 1 2 satisfy x 2 2 x 1 0. 99. Both 1 i and 1 i satisfy x 2 2 x 2 0. 100. Both 2 3 i and 2 3 i satisfy x 2 4 x 13 0. Solve each problem. 101. Approach speed. The formula 1211.1 L CA 2 S is used to determine the approach speed for landing an aircraft, where L is the gross weight of the aircraft in pounds, C is the coefficient of lift, S is the surface area of the wings in square feet (ft 2 ), and A is approach speed in feet per second. Find A for the Piper Cheyenne, which has a gross weight of 8700 lb, a coefficient of lift of 2.81, and wing surface area of 200 ft 2 . dug22241_ch10a.qxd 11/10/2004 18:30 Page 628
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10-13 10.2 The Quadratic Formula 629 102. Time to swing. The period T (time in seconds for one com- plete cycle) of a simple pendulum is related to the length L (in feet) of the pendulum by the formula 8 T 2 2 L . If a child is on a swing with a 10-foot chain, then how long does it take to complete one cycle of the swing? 103. Time for a swim. Tropical Pools figures that its monthly revenue in dollars on the sale of x aboveground pools is given by R 1500 x 3 x 2 , where x is less than 25. What number of pools sold would provide a revenue of $17,568? 104. Pole vaulting. In 1981 Vladimir Poliakov (USSR) set a world record of 19 ft 3 4 in. for the pole vault ( ). To reach that height, Poliakov obtained a speed of approximately 36 feet per second on the runway. The formula h 16 t 2 36 t gives his height t seconds after leaving the ground. a) Use the formula to find the exact values of t for which his height was 18 feet. b) Use the accompanying graph to estimate the value of t for which he was at his maximum height. c) Approximately how long was he in the air? Getting More Involved 105. Discussion Which of the following equations is not a quadratic equation? Explain your answer. a) x 2 5 x 1 0 b) 3 x 2 1 0 c) 4 x 5 0 d) 0.009 x 2 0 106. Exploration Solve x 2 4 x k 0 for k 0, 4, 5, and 10. a) When does the equation have only one solution? b) For what values of k are the solutions real? c) For what values of k are the solutions imaginary? 107. Cooperative learning Write a quadratic equation of each of the following types, then trade your equations with those of a classmate. Solve the equations and verify that they are of the required types.
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