redch0 - Contents 0 REVIEW NOTES 1 0.0 Quadratic Equations...

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Contents 0 REVIEW NOTES 1 0.0 Quadratic Equations and Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.3 Absolute Values and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 0.4 Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 0.5 The Chart Method for Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 0.6 Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 0.7 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 0.7.1 Radian Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 0.7.2 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 0.7.3 Trigonometric Identities and Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 0.7.4 Law of Cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 0.8 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
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2 CONTENTS
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Chapter 0 REVIEW NOTES 0.0 Quadratic Equations and Completing the Square Factor the following quadratic expressions 1. x 2 - 3 x , 2 x 2 + x , πx 2 + 3 π , 17 x 2 + 51 x and 2 x 2 + 2 x 2. x 2 - 9 and x 2 - 49 3. x 2 - 3 and x 2 - 1 4. x 2 - 2 and x 2 - π 5. x 2 + x + 1 / 4 and x 2 + x + 6 6. x 2 - x + 1 / 4 and x 2 - x - 6 7. x 2 + 2 a + a 2 and x 2 + ( a + b ) x + ab 8. x 2 - 2 x + 3 and x 2 - 3 x - 40 9. 3 x 2 - 5 x - 2 and 8 x 2 + 2 x - 1 10. 2 x 2 - 5 bx - 3 b 2 and x 2 - 4 x - 21 11. x 4 - 2 x 2 + 1 and x 6 - 4 x 3 - 21 Solve the following equations by completing the square 12. 3 x 2 + 6 x - 1 = 0 13. 3 x (3 x - 2) = 6 x - 5 14. y 2 - 15 y - 4 = 0 15. 6 u 2 + 7 u - 3 = 0 16. x 2 - 2 x + 9 = 0 17. 4 z 2 - 4 z - 1 = 0 18. p (2 p - 4) = 5 19. ( x - 2) 2 + 3 x - 5 = 0 20. (3 x - 2) 2 + ( x + 1) 2 - 0 21. 5 y 2 - 15 y + 9 = 0 Use the quadratic formula to solve the following equations 22. 5 x 2 + 6 x - 1 = 0 23. 2 x 2 = 18 x + 5 24. x (2 x - 3) = 2 x - 6 25. 6 x 2 - 7 x + 2 = 0 26. 2 x 2 = 13( x - 1) + 3 27. 2 x 2 - 6 x - 1 = 0 28. 1200 y 2 = 10 y + 1 29. x 2 + 2 bx - c 2 = 0 30. x 2 - 6 ax + 3 a 2 = 0 31. πu 2 + ( π 2 - 1) u - π = 0 32. x ( x - 2 + 4) = 4( x + 1) 33. 3 x 2 = 5( x - 1) 2
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2 REVIEW NOTES For the following functions f find the discriminant of f ( x ) = 0 and determine whether the roots are real and unequal, real and equal, or do not exist. Graph f ( x ) without plotting more than four points. 34. f ( x ) = 4 x 2 - 4 x + 1 35. f ( z ) = z 2 + z + 1 36. f ( x ) = 4 x 2 - x - 5 37. f ( x ) = 7 x - 5 x - 2 38. f ( x ) = x 2 + 2 x + 1 / 4 39. f ( x ) = x 2 - ax - 1 40. f ( x ) = 3 x 2 + πx + 4 41. f ( x ) = x 2 - 2 ax + a 2 42. f ( x ) = 3 x 2 - 2 x - 3 43. f ( x ) = 9 x 2 - 12 x + 4 Determine the values of K for which the equation has real and equal roots. 44. 5 x 2 - 4 x - (5 + K ) = 0 45. ( K + 2) x 2 + 3 x + ( K + 3) = 0 46. x 2 + 3 - K (2 x - 2) = 0 47. ( K + 2) x 2 + 5 Kx - 2 = 0 48. x 2 - x (2 + 3 K ) + 7 = 0 49. ( K - 1) x 2 + 2 x + ( K + 1) = 0 Complete the square for each of the following functions Examples : a) p ( x ) = x 2 + 2 x + 10 b) p ( x ) = x 2 - x p ( x ) = x 2 + 2 x + 1 - 1 + 10 p ( x ) = x 2 - 2 ( 1 / 2 ) x + ( 1 / 2 ) 2 - ( 1 / 2 ) 2 p ( x ) = ( x + 1) 2 + 9 p ( x ) = x 2 - x + 1 / 4 - 1 / 4 p ( x ) = ( x - 1 / 2 ) 2 - 1 / 4 . 50. p ( x ) = x 2 + 5 x + 2 51. p ( x ) = x 2 + 3 x + 1 52. p ( t ) = - 3 t 2 - 5 t + 1 53. p ( t ) = t 2 - 2 t 54. p ( x ) = x 2 + 3 x 55. p ( x ) = x 2 + 4 x - 3 56. p ( x ) = 4 x 2 + 12 x + 10 57. p ( x ) = - 16 x 2 + 6 x 58. p ( x ) = x 2 + 4 b + c 59. p ( x ) = ax 2 + ax + b 60. p ( x ) = π ( x 2 - 2 x ) 61. p ( x ) = 24( - x - 3 x + 1) In the following questions write the given expression in the form a p c 2 ± ( dx + b ) 2 and simplify where c is a fraction. Example : - 4 x 2 + x
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0.1 Polynomials 3 Let q ( x ) = - 4 x 2 + x q ( x ) = - 4 ( x 2 - x 4 ) complete square next: q ( x ) = - 4 x 2 - 2 ( 1 8 ) x + ( 1 8 ) 2 - ( 1 8 ) 2 q ( x ) = - 4 ( x - 1 8 ) 2 - 1 64 . Since - 4 = ( - 1)4 we get q ( x ) = 4 1 64 - ( x - 1 8 ) 2 p q ( x ) = 2 q 1 64 - ( x - 1 8 ) 2 = 2 r 1 - 64 ( x - 1 8 ) 2 64 p q ( x ) = 1 / 4 q 1 - 64 ( x - 1 8 ) 2 = 1 / 4 q 1 - ( 8 ( x - 1 8 )) 2 p q ( x ) = 1 / 4 p 1 - (8 x - 1) 2 62. x 2 + 3 x + 1 63. - 3 t 2 + 5 t + 1 64. x 2 + 3 x 65. - 16 x 2 + 6 x 66. - x 2 + x + 1 67.
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