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Algebra 1 Solving Quadratic Equations Name _____________________________________ Per _____ Lesson 9.4: You can solve by graphing. The solutions of a quadratic equation and the related x-intercepts are often called the roots of the equation or zeros of the function. They are where the parabola crosses the x-axis (x-intercepts). Leave in simplest radical form, if necessary. You can solve equations of the form x2 = a by finding the square roots of each side. 1. 2x2 - 98 = 0 2. t2 - 25 = 0 3. 3n2 + 12 = 12 4. 2g2 + 32 = 0. 5. x2 - 9 = 0 9. 1 2 x 3 -3=0 13. 27 - y2 = 0 6. x2 + 5 = 0 7. 2x2 - 8 = 0 10. x2 - 72 = 0 11. 14. c2 + 25 = 25 15. x2 - 9 = -16 17. Find the side of a square with an area of 256 m2. 1 2 x 4 -1=0 8. 4x2 = 0 12. k2 = 49 16. 6x2 - 24 = 0 18. Find the side of a square with an area of 90 ft2. Lesson 9.5 Solve by Factoring The Zero-Product Property: For every real number a and b, if ab = 0, then either a = 0 or b = 0. If (x + 3)(x + 2) = 0, then either x + 3 = 0 or x + 2 = 0. Make sure the quadratic equations are in standard form! If possible, factor GCF first! 19. x2 + 6x + 8 = 0 20. x2 - 8x - 48 = 0 21. 2x2 - 5x = 88 22. x2 - 12x = -36 23. x2 - 6x - 27 = 0 24. 2x2 - x - 36 = 0 25. t2 + t = 0 26. n2 + n - 12 = 0 27. k3 + 7k2 = -12k 28. b2 + 3b - 4 = 0 29. w2 - 8w = 0 30. m2 - 5m - 14 = 0 31. 2q2 + 22q = -60 32. x2 + 8x = -15 33. 3q2 + 16q = -5 34. t2 - 3t = 28 35. 4y2 = 25 36. k2 - 3k - 10 = 0 37. 4 = -5n + 6n2 38. 2x2 - 7x = 15 39. p3 - 2p2 - 48p = 0 40. 3t2 + 8t = t2 - 3t - 12 41. 4x2 + 20 = 10x + 3x2 - 4 42. 2c2 - 7c = -5 Lesson 9.6 You can also solve by Completing the Square. To find the number to add to each side of the equation 𝒃 𝟐 x2 + bx = c to complete the square, you must use the formula (𝟐) . What number would you add to each equation to make a perfect square trinomial? (No coefficent can be on x2) 43. k2 + 14k = 0 44. r2 + 8r = 48 45. x2 + 10x + 17 = 0 46. p2 - 12p + 11 = 0 47. 2k2 + 5k = k + 10 Lesson 9.7 Solve equations ax2 + bx + c = 0 by using the quadratic formula x = −𝒃 ± √𝒃𝟐 −𝟒𝒂𝒄 . 𝟐𝒂 Make sure you simplify all radicals! Make sure all equations are in standard form. 48. x + 6 - 5x = 0 49. x2 - 2x - 8 = 0 50. 2x2 + 5x + 3 = 0 51. 3x2 - 10x = 2 2 52. x2 + 2x = 13 53. 5x2 + 13x - 1 = 0 54. 3x2 - 4x = 2 55. 8x2 - 2x = 7 56. x2 + x = 5 57. x2 + 2x - 7 = 0 58. 2x2 - 7x = -6 59. x2 - 4x = -10 You can use the discriminant to determine the # of solutions of the equation ax2 + bx + c = 0 (x-intercepts, roots of the equation, zeros of the function). The discriminant is b2 - 4ac. If b2 - 4ac > 0, there are 2 solution; if b2 - 4ac = 0, there is 1 solution; if b2 - 4ac < 0 there are no solutions. Determine the # of solutions. DO NOT SOLVE! Make sure the equations are in standard form! 60. x2 - 6x + 3 = 0 61. x2 - 6x + 9 = 0 62. 3x2 - 5x - 1 = 0 63. x2 - 6x = -12 64. 3x2 - 4x = 7 Lesson 9.8 Mixed Review: Solve the following quadratic equations using any method you choose. If there is no solution, write "no solution." 65. 4x2 - 81 = 0 66. 3x2 - 5x + 9 = 0 67. x2 + 13x = -36 68. x2 = 100 69. 2t2 = 72 70. 5b2 - 10 = 0 71. 3m2 - 6m = 2 72. 2r2 - 24r + 72 = 0 73. x2 + 2x - 1 = 0 74. x2 - 8x = 33 75. x3 - 5x2 + 4x = 0 76. x2 + 64 = 0 77. 2c2 + 11c + 15 = 0 78. x3 - x2 = 0 79. x2 + 1 = 0 80. Which equation does a. -9x + 5x = -7 9 ± √(−9)2 − 4(5)(−7) 2(5) b. 5x2 - 9x = -7 give the solutions to? c. 5x2 + 7x = 9 d. 5x2 - 9x = 7