6.9 Solving Quadratic Equations by Using Factoring and by Using the Quadratic Formula 385 Exercises Warm Up Exercises In Exercises 1– 6, fill in the blank with an appropriate word, phrase, or symbol(s). 1. An expression that contains two terms in which each exponent that appears on a variable is a whole number is called a(n) ________. Binomial 2. An expression that contains three terms in which each exponent that appears on a variable is a whole number is called a(n) ________. Trinomial 3. When multiplying two binomials, a method that obtains the products of the first, outer, inner, and last terms is called the ________ method. FOIL 4. The property that indicates that if the product of two factors is 0, then one or both of the factors must have a value of 0 is called the ________ property. Zero-factor 5. When a quadratic equation cannot be easily solved by factoring, the equation can be solved using the ________ formula. Quadratic 6. For a quadratic equation in standard form ax bx c a 0, 0, 2 + + = ≠ the quadratic formula is _______. x b b ac a 4 2 2 = − ± − Practice the Skills In Exercises 7–12, multiply the binomials. 7. x x ( 3)( 11) + − x x8 33 2 − − 8. x x ( 7)( 5) − + x x2 35 2 − − 9. x x (2 3)(3 1) + − x x 6 7 3 2 + − 10. x x (3 2)(2 1) − + x x 6 2 2 − − 11. x x (6 7)(8 9) − + x x 48 2 63 2 − − 12. x x (7 9)(6 5) + − x x 42 19 45 2 + − In Exercises 13–24, factor the trinomial. 13. x x5 6 2 + + x x ( 2)( 3) + + 14. x x7 10 2 + + x x ( 2)( 5) + + 15. x x 6 2 − − x x ( 3)( 2) − + 16. x x3 10 2 − − x x ( 5)( 2) − + 17. x x3 10 2 + − x x ( 5)( 2) + − 18. x x5 14 2 + − x x ( 7)( 2) + − 19. x x2 3 2 − − x x ( 1)( 3) + − 20. x x4 5 2 − − x x ( 1)( 5) + − 21. x x9 18 2 − + x x ( 6)( 3) − − 22. x x6 8 2 − + x x ( 4)( 2) − − 23. x x3 28 2 + − x x ( 7)( 4) + − 24. x x4 32 2 + − x x ( 4)( 8) − + In Exercises 25–36, factor the trinomial. 25. x x 2 7 3 2 + + x x (2 1)( 3) + + 26. x x 3 7 2 2 + + x x (3 1)( 2) + + 27. x x 3 2 2 + − x x (3 2)( 1) − + 28. x x 2 3 2 − − x x (2 3)( 1) − + 29. x x 3 17 10 2 − + x x (3 2)( 5) − − 30. x x 2 9 10 2 − + x x (2 5)( 2) − − 31. x x 4 11 6 2 + + x x (4 3)( 2) + + 32. x x 4 20 21 2 + + x x (2 7)(2 3) + + 33. x x 4 11 6 2 − + x x (4 3)( 2) − − 34. x x 6 11 4 2 − + x x (3 4)(2 1) − − 35. x x 8 2 3 2 − − x x (4 3)(2 1) − + 36. x x 6 13 2 2 − + x x (6 1)( 2) − − In Exercises 37– 40, solve each equation, using the zero-factor property. 37. x x ( 6)(5 4) 0 − − = 6, 4 5 38. x x (2 3)( 1) 0 + − = 3 2 , 1 − 39. x x (3 4)(2 1) 0 + − = 4 3 , 1 2 − 40. x x (5 4)(3 2) 0 − + = − 4 5 , 2 3 In Exercises 41–58, solve each equation by factoring. 41. x x8 15 0 2 + + = − − 5, 3 42. x x9 14 0 2 + + = 7, 2 − − 43. x x8 7 0 2 − + = 7, 1 44. x x 12 11 0 2 − + = 11, 1 45. x x 15 2 2 − = − 5, 3 46. x x 21 4 2 − = 3, 7 − 47. x x4 3 2 = − 3, 1 48. x x8 7 2 = − 1, 7 49. x x 6 19 15 0 2 + + = 3 2 , 5 3 − − 50. x x 6 25 14 0 2 + + = 2 3 , 7 2 − − 51. x x 3 10 8 2 + = − 2 3 , 4 52. x x 2 15 27 2 + = 9, 3 2 − 53. x x 3 5 2 2 = − + − 1 3 , 2 54. x x 2 5 3 2 = − + − 1 2 , 3 55. x x 3 5 2 2 = − − − − 2 3 , 1 56. x x 2 5 3 2 = − − − − 3 2 , 1 57. x x 20 15 13 2 = − 5 4 , 3 5 − 58. x x 15 16 8 2 = − 4 3 , 4 5 − In Exercises 59–78, solve the equation, using the quadratic formula. If the equation has no real solution, so state. 59. x x2 3 0 2 + − = 3, 1 − 60. x x 11 24 0 2 − + = 8, 3 61. x x3 18 0 2 − − = − 6, 3 62. x x6 16 0 2 − − = − 8, 2 SECTION 6.9
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