Math Midterm Chapter 1 Solution Key

Advanced Math Midterm Review Chapter 1 Name: Solve the following equations. Check your solutions. 1. 6(x-2) - 2(y-7) = 0 3x - y = 10 2. 6(x-1.5) = 12g 6x - 9 = 12g 9 = 6g g = 3/2 3. (3/4)x + 1/2 = 2/3 9x + 6 = 8 9x = 2 x = 2/9 4. 18(4/9x - 1/2) = 2/3 8x - 9 = 12 8x = 21 x = 21/8 5. |(x-4)/2| = 8 |x-4| = 16 x-4 = 16 or x-4 = -16 x = 20 or x = -12 6. |x-6| = 9 x-6 = 9 or x-6 = -9 x = 15 or x = -3 7. 3|x+2| - 16 = 2 3|x+2| = 18 |x+2| = 6 x+2 = 6 or x+2 = -6 x = 4 or x = -8 8. (3-2i) + (2+5i) = 5+3i 9. (2-i)² = (2-i)(2-i) = 4 - 2i - 2i + i² = 3-4i 10. i⁴ = (i²)² = (-1)² = 1 11. 402 ÷ 4 = 100 R2 12. √-36 = i√36 = 6i 13. √-98 = √-49 * √2 = 7i√2 14. √-55 = i√55 15. (2-5i) + (6-3i) = 8-8i 16. (3-2i)(2+5i) = 6 + 15i - 4i - 10i² = 16 + 11i 17. (2-i)² = (2-i)(2-i) = 4 - 2i - 2i + i² = 3 - 4i 18. 3i(6-4i) = 18i - 12i² = 12 + 18i 19. (2-3i)(4-5i) = 8 - 10i - 12i + 15i² = -7 - 22i 20. (2-4i)/(1+4i) * (1-4i)/(1-4i) = (2-8i-4i+16i²)/(1-16i²) = (-14-12i)/17 21. (3-i)/(2-5i) * (2+5i)/(2+5i) = (6+15i-2i-5i²)/(4+10i-10i-25i²) = (11+13i)/29 22. 2/i * i/i = 2i/i² = -2i 23. 3/0i * 2/2 = 6i/0i² = -3i/8 Ch. 1 Given the graph of the quadratic equation, estimate the solutions (zeros/roots). 24. -3 and 3 25. Between -1 and 0, and between 0 and 1 26. 2 and 4 Estimate the solutions (zeros/roots) for the following equations. Then use the zeros/roots to graph. 27. y = -(x+2)(x+8) x = -2, -8 opens down 28. y = x² + 3x + 2 y = (x+1)(x+2) x = -1, -2 opens up 29. y = 5x² - 35x + 50 y = 5(x²-7x+10) y = 5(x-2)(x-5) x = 2, 5 opens up Solve the following quadratics by factoring. 30. 4x² - 2x = 0 2x(2x-1) = 0 2x = 0 or 2x-1 = 0 x = 0, x = 1/2 31. 2x² - 8 = 0 2(x²-4) = 0 2(x+2)(x-2) = 0 x = -2, x = 2 32. x² - 144 = 0 (x+12)(x-12) = 0 x = -12, x = 12 33. x²+x = 30 x²+x-30 = 0 (x+6)(x-5) = 0 x = -6, x = 5 34. x² + 16 = 8x x²-8x+16 = 0 (x-4)(x-4) = 0 x = 4 35. 12x²-25x+12=0 (4x-3)(3x-4) = 0 4x-3 = 0 or 3x-4 = 0 x = ¾, x = 4/3 36. 36x²-12x+1=0 (6x-1)(6x-1) = 0 6x-1 = 0 x = 1/6 37. 2x²+4x=6 2x²+4x-6=0 x²+2x-3=0 (x+3)(x-1)=0 x = -3, x = 1 38. 12x²-22x-14=0 6x²-11x-7=0 (2x+1)(3x-7)=0 x = -1/2, x = 7/3 Ch. 1 Solve the following equations that are in quadratic form. 39. x⁴ - 29x² + 100 = 0 (x²-25)(x²-4) = 0 (x+5)(x-5)(x+2)(x-2) = 0 x = -5, 5, -2, 2 40. x⁴ + 17x² + 16 = 0 (x²+16)(x²+1) = 0 (x+4i)(x-4i)(x+i)(x-i) = 0 x = -4i, 4i, -i, i Solve the following quadratics using square roots. 41. (x+2)² = 25 x+2 = ±5 x = -2 ± 5 x = 3, -7 42. (x-6)² = 81 x-6 = ±9 x = 6 ± 9 x = 15, -3 43. (x-5)² = -32 x-5 = ±√-32 x-5 = ±4i√2 x = 5 ± 4i√2 Solve the following quadratics by completing the square. 44. x² - 8x = -14 x² - 8x + 16 = -14 + 16 (x-4)² = 2 x-4 = ±√2 x = 4 ± √2 45. x² + 6x - 10 = 0 x² + 6x + 9 = 10 + 9 (x+3)² = 19 x+3 = ±√19 x = -3 ± √19 46. x² - 3x - 5 = 0 x² - 3x + 9/4 = 5 + 9/4 (x-3/2)² = 29/4 x-3/2 = ±√29/2 x = 3/2 ± √29/2 47. x² + 12x - 4 = 0 x² + 12x + 36 = 4 + 36 (x+6)² = 40 x+6 = ±2√10 x = -6 ± 2√10 48. x² + 16x + 68 = 0 x² + 16x + 64 = -68 + 64 (x+8)² = -4 x+8 = ±2i x = -8 ± 2i 49. x² - 2x + 8 = 0 x² - 2x + 1 = -8 + 1 (x-1)² = -7 x-1 = ±i√7 x = 1 ± i√7 Find the value of the discriminant. Describe the type of roots for the following quadratics (how many and what type). 50. x² - 10x + 16 = 0 b²-4ac = (-10)² - 4(1)(16) = 36 2 real rational roots 51. x² + 6x + 12 = 0 b²-4ac = 6² - 4(1)(12) = -12 2 imaginary conjugates 52. 12x² + 15x + 7 = 0 (15)² - 4(12)(7) = -111 2 imaginary conjugates Ch. 1 Solve the following quadratics using the quadratic formula. 53. 2x² - 5x + 1 = 0 x = (5 ± √(5²-4*2*1))/(2*2) x = (5 ± √17)/4 54. x² + 6x + 9 = 0 x = (-6 ± √(6²-4*1*9))/(2*1) x = -6/2 = -3 55. 3x² + 6x + 7 = 0 x = (-6 ± √(6²-4*3*7))/(2*3) x = (-6 ± √-36)/6 x = (-6 ± 6i)/6 x = -1 ± i Solve each equation. Be sure to check your solution(s). 56. x^(3/2) = 10 (x^(3/2))^(2/3) = 10^(2/3) x = 100^(1/3) 57. x^(5/6) = 32 (x^(5/6))^(6/5) = 32^(6/5) x = (2^5)^(6/5) x = 2^6 x = 64 58. 3x^(2/3) = 75 x^(2/3) = 25 x = 25^(3/2) x = (5²)^(3/2) x = 125 59. 7x^(2/3) = 56 x^(2/3) = 8 x = 8^(3/2) x = (2³)^(3/2) x = 2^9 x = 512 60. 2x^(2/3) + 6 = 60 2x^(2/3) = 54 x^(2/3) = 27 x = 27^(3/2) x = (3³)^(3/2) x = 3^9 x = 19683 61. (x²-9)(x²-14) = 0 x² - 9 = 0 or x² - 14 = 0 x² = 9 or x² = 14 x = ±3 or x = ±√14 62. 4x^(2/3) - 54 = 270 4x^(2/3) = 324 x^(2/3) = 81 x = 81^(3/2) x = (3⁴)^(3/2) x = 3^6 x = 729