chapter two Equations and Inequalities of One Variable ho = 50 ft Vo = 60 ft/s 50 ft o = -16t2 + 60t +' 50 The 0= 8t2 - 30t ~ 25 good method t 30 ± ..}900 + 800 = ---16 30 ± 1O.Ji7 t=--16 quadratic polynomial in the equation factor over the integers. so the quadratic is a to use. To simplify the calculations, we can begin by dividing both sides of the equation by -2. We then proceed to reduce the radical and simplify the resulting fraction (which actuaHy describes two solutions ). 15 ± 5.Ji7 t=--- These numbers t",,~,4.45 The negative solution is immaterial in this problem, as this represents a time before the ball is thrown, so it 8 is discarded. thrown. are most meaningful 3x2 4x2+ -7x -339 =2X2 = 0 +18 9x-5x2 -3x= =-2x2 + 14x 1. 2X2 2x2-x=3 132 in decimal form. The ball lands 4.45 seconds Solve the following quadratic equations by factoring. See Example 1. 3. 9. 5x2 3 = 4x2 + 6x -1 5. y(2y+ 9)+=-9 7. x2-14x+49=0 (3x+2)(x-l)=7-7x 11.+ 2x (x-7)2=16 doesn't formula 6. 2. 4. 15x2 +x = 2 ..10. 12. 8. after being section 2.3 Quadratic Equations in One Variable x2-6x+9=-16 2X2 u2+10u+9=0 +6x-10 10 33. Z2+26z+2 = =-23 21. x2-4x+4=49 27. 29. 2x2+7x-1S=0 32. 26. 30. 4x2 -56x++9=0 195 0 14. (a-2(=-S -1=0 23. 17. (3x-6)2 =4x2 y2+22y+96=0 Solve (Y_18)2 the following-quadratic equations the= square root method. See Example 2. 15. 20. 0by 24. (2x+3)2 18. 9(8t-3)2 = (3s+ ~2)2 0 25. See x2 +Example 8x += 79 = -8 13. (X_3)2 dratic equations by completing the square. 3. Solve the follow~ng quadratic equations by using the quadratic formula. See Example 4. 2X2 3x2 +8x-3 6x==-1 6x0=- 30 50. 3x2-4=-x 7x2 3a2 -42x +--5x-lO:;= -4x 12a-576 0=lOz 4Z2 14z == 38. 57. 44. 56. 34. 7x2 y2 l+24y+23=0 4x2 -2y+1 -3x =-1 -289 53. 42. 51. 41. 4x2-14x-27=3 x2 ==27 47. 59. 39. 3x2 0 -768 5x2 ·(z-11)2 a(a+ (y_8)2 2) === 36 9051 45. 35. x2 -6x +-2x 20x+36 =-48 36. 2.1l-3.Sy=4 54. -3(b+ 5)2 48. (9Y_6)2 = =121y2 Solve the following quadratic equations using any appropriate method. 20 60. 4w2 + 10w+5 = 3w2 + 18w-1O 43. l +9y = -40.50 / 133 chapter two Equations and Inequalities of One Variable Solve the following application problems. See Example 5. 61. How long would it take for a ball dropped from the top of a 144-foot building to hit the ground? 62. Suppose that instead of being dropped, as in problem 61, a ball is thrown upward with a velocity of 40 feet per second from the top of a 144-foot building. Assuming it misses the building on the way back down, how long after being thrown will it hit the ground? = 144 ft Vo = 40 ft/sec ho 144 ft 63. A slingshot is used to shoot a BB at a velocity of 96 feet per second straight up from ground level. When will the BB r~ach its maximum height of 144 feet? 64. A rock is thrown upward with a velocity of 20 meters per second from the top of a 24 meter high cliff, and it misses the cliff on the way back down. When will the rock be 7 -meters from ground level? (Round your answer to the nearest tenth.) 134 section2~3 . Quadratic Equations in One Variable 65. Luke, an experienced bungee jumper, leaps from· a tall bridge and falls toward the river below. The bridge is 170 feet above the water and Luke's bungee cord is 110 feet long unstretched. When will Luke's cord begin to stretch? (Round your answer to the nearest tenth.) Use the connection between solutions of quadratic equations and polynomial factoring to answer the following questions. See the discussion after Example 3. 66. Factor the quadratic x2 67. Factor thy quadratic 9x2 6x + 13. - - 6x - 4. 68. Factor the quadratic 4x2 + 12x + 1. 69. Factor the quadratic 25x2 -lOx + 2. 70. Determine band c so that the equation x2 + bx + c = 0 has the solution set {-3,8}. 135
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