51. x2 - x - 2 = 0 52. x²-2x-3=0 53. x²-3x - 4 = 0 54. x² - 4x - 5=0 55. x²- 5x - 6=0 56. x2 - x - 6 = 0 57. x²-7 x 8 = 0 58. x²-2x-8 = 0 59. x²-9x-10 = 0 60. x²-3x 10 = 0 61. x²-x-30 = 0 62. x²- 26x - 27 = 0 63. x²-6x - 27 = 0 x²-35x36=0 64. x² - 35x - 36 = 65. x² - 16x - 36 = 0 66. x²- 9x - 36 = 0 67. x²- 5x-36 = 0 68. x² - 39x - 40 = 0 69. x²-18x - 40 = 0 70. x²-6x - 40 = 0 71. x²-3 x - 40 = 0 72. x²-11x- 26 = 0 73. x²-3x-54 = 0 74. x²-12 x - 64 = 0 75. x²-2 x - 80 = 0

{ "task": "Решить квадратные уравнения.", "solution": [ { "number": 51, "equation": "x² - x - 2 = 0", "solution": "Для решения квадратного уравнения вида ax² + bx + c = 0, используем формулу дискриминанта D = b² - 4ac. Затем находим корни x = (-b ± √D) / (2a).В данном случае a = 1, b = -1, c = -2.D = (-1)² - 4 * 1 * (-2) = 1 + 8 = 9.√D = √9 = 3.x₁ = (1 + 3) / (2 * 1) = 4 / 2 = 2.x₂ = (1 - 3) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 2, x₂ = -1" }, { "number": 52, "equation": "x² - 2x - 3 = 0", "solution": "a = 1, b = -2, c = -3.D = (-2)² - 4 * 1 * (-3) = 4 + 12 = 16.√D = √16 = 4.x₁ = (2 + 4) / (2 * 1) = 6 / 2 = 3.x₂ = (2 - 4) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 3, x₂ = -1" }, { "number": 53, "equation": "x² - 3x - 4 = 0", "solution": "a = 1, b = -3, c = -4.D = (-3)² - 4 * 1 * (-4) = 9 + 16 = 25.√D = √25 = 5.x₁ = (3 + 5) / (2 * 1) = 8 / 2 = 4.x₂ = (3 - 5) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 4, x₂ = -1" }, { "number": 54, "equation": "x² - 4x - 5 = 0", "solution": "a = 1, b = -4, c = -5.D = (-4)² - 4 * 1 * (-5) = 16 + 20 = 36.√D = √36 = 6.x₁ = (4 + 6) / (2 * 1) = 10 / 2 = 5.x₂ = (4 - 6) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 5, x₂ = -1" }, { "number": 55, "equation": "x² - 5x - 6 = 0", "solution": "a = 1, b = -5, c = -6.D = (-5)² - 4 * 1 * (-6) = 25 + 24 = 49.√D = √49 = 7.x₁ = (5 + 7) / (2 * 1) = 12 / 2 = 6.x₂ = (5 - 7) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 6, x₂ = -1" }, { "number": 56, "equation": "x² - x - 6 = 0", "solution": "a = 1, b = -1, c = -6.D = (-1)² - 4 * 1 * (-6) = 1 + 24 = 25.√D = √25 = 5.x₁ = (1 + 5) / (2 * 1) = 6 / 2 = 3.x₂ = (1 - 5) / (2 * 1) = -4 / 2 = -2.", "roots": "x₁ = 3, x₂ = -2" }, { "number": 57, "equation": "x² - 7x - 8 = 0", "solution": "a = 1, b = -7, c = -8.D = (-7)² - 4 * 1 * (-8) = 49 + 32 = 81.√D = √81 = 9.x₁ = (7 + 9) / (2 * 1) = 16 / 2 = 8.x₂ = (7 - 9) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 8, x₂ = -1" }, { "number": 58, "equation": "x² - 2x - 8 = 0", "solution": "a = 1, b = -2, c = -8.D = (-2)² - 4 * 1 * (-8) = 4 + 32 = 36.√D = √36 = 6.x₁ = (2 + 6) / (2 * 1) = 8 / 2 = 4.x₂ = (2 - 6) / (2 * 1) = -4 / 2 = -2.", "roots": "x₁ = 4, x₂ = -2" }, { "number": 59, "equation": "x² - 9x - 10 = 0", "solution": "a = 1, b = -9, c = -10.D = (-9)² - 4 * 1 * (-10) = 81 + 40 = 121.√D = √121 = 11.x₁ = (9 + 11) / (2 * 1) = 20 / 2 = 10.x₂ = (9 - 11) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 10, x₂ = -1" }, { "number": 60, "equation": "x² - 3x - 10 = 0", "solution": "a = 1, b = -3, c = -10.D = (-3)² - 4 * 1 * (-10) = 9 + 40 = 49.√D = √49 = 7.x₁ = (3 + 7) / (2 * 1) = 10 / 2 = 5.x₂ = (3 - 7) / (2 * 1) = -4 / 2 = -2.", "roots": "x₁ = 5, x₂ = -2" }, { "number": 61, "equation": "x² - x - 30 = 0", "solution": "a = 1, b = -1, c = -30.D = (-1)² - 4 * 1 * (-30) = 1 + 120 = 121.√D = √121 = 11.x₁ = (1 + 11) / (2 * 1) = 12 / 2 = 6.x₂ = (1 - 11) / (2 * 1) = -10 / 2 = -5.", "roots": "x₁ = 6, x₂ = -5" }, { "number": 62, "equation": "x² - 26x - 27 = 0", "solution": "a = 1, b = -26, c = -27.D = (-26)² - 4 * 1 * (-27) = 676 + 108 = 784.√D = √784 = 28.x₁ = (26 + 28) / (2 * 1) = 54 / 2 = 27.x₂ = (26 - 28) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 27, x₂ = -1" }, { "number": 63, "equation": "x² - 6x - 27 = 0", "solution": "a = 1, b = -6, c = -27.D = (-6)² - 4 * 1 * (-27) = 36 + 108 = 144.√D = √144 = 12.x₁ = (6 + 12) / (2 * 1) = 18 / 2 = 9.x₂ = (6 - 12) / (2 * 1) = -6 / 2 = -3.", "roots": "x₁ = 9, x₂ = -3" }, { "number": 64, "equation": "x² - 35x - 36 = 0", "solution": "a = 1, b = -35, c = -36.D = (-35)² - 4 * 1 * (-36) = 1225 + 144 = 1369.√D = √1369 = 37.x₁ = (35 + 37) / (2 * 1) = 72 / 2 = 36.x₂ = (35 - 37) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 36, x₂ = -1" }, { "number": 65, "equation": "x² - 16x - 36 = 0", "solution": "a = 1, b = -16, c = -36.D = (-16)² - 4 * 1 * (-36) = 256 + 144 = 400.√D = √400 = 20.x₁ = (16 + 20) / (2 * 1) = 36 / 2 = 18.x₂ = (16 - 20) / (2 * 1) = -4 / 2 = -2.", "roots": "x₁ = 18, x₂ = -2" }, { "number": 66, "equation": "x² - 9x - 36 = 0", "solution": "a = 1, b = -9, c = -36.D = (-9)² - 4 * 1 * (-36) = 81 + 144 = 225.√D = √225 = 15.x₁ = (9 + 15) / (2 * 1) = 24 / 2 = 12.x₂ = (9 - 15) / (2 * 1) = -6 / 2 = -3.", "roots": "x₁ = 12, x₂ = -3" }, { "number": 67, "equation": "x² - 5x - 36 = 0", "solution": "a = 1, b = -5, c = -36.D = (-5)² - 4 * 1 * (-36) = 25 + 144 = 169.√D = √169 = 13.x₁ = (5 + 13) / (2 * 1) = 18 / 2 = 9.x₂ = (5 - 13) / (2 * 1) = -8 / 2 = -4.", "roots": "x₁ = 9, x₂ = -4" }, { "number": 68, "equation": "x² - 39x - 40 = 0", "solution": "a = 1, b = -39, c = -40.D = (-39)² - 4 * 1 * (-40) = 1521 + 160 = 1681.√D = √1681 = 41.x₁ = (39 + 41) / (2 * 1) = 80 / 2 = 40.x₂ = (39 - 41) / (2 * 1) = -2 / 2 = -1.", "roots": "x₁ = 40, x₂ = -1" }, { "number": 69, "equation": "x² - 18x - 40 = 0", "solution": "a = 1, b = -18, c = -40.D = (-18)² - 4 * 1 * (-40) = 324 + 160 = 484.√D = √484 = 22.x₁ = (18 + 22) / (2 * 1) = 40 / 2 = 20.x₂ = (18 - 22) / (2 * 1) = -4 / 2 = -2.", "roots": "x₁ = 20, x₂ = -2" }, { "number": 70, "equation": "x² - 6x - 40 = 0", "solution": "a = 1, b = -6, c = -40.D = (-6)² - 4 * 1 * (-40) = 36 + 160 = 196.√D = √196 = 14.x₁ = (6 + 14) / (2 * 1) = 20 / 2 = 10.x₂ = (6 - 14) / (2 * 1) = -8 / 2 = -4.", "roots": "x₁ = 10, x₂ = -4" }, { "number": 71, "equation": "x² - 3x - 40 = 0", "solution": "a = 1, b = -3, c = -40.D = (-3)² - 4 * 1 * (-40) = 9 + 160 = 169.√D = √169 = 13.x₁ = (3 + 13) / (2 * 1) = 16 / 2 = 8.x₂ = (3 - 13) / (2 * 1) = -10 / 2 = -5.", "roots": "x₁ = 8, x₂ = -5" }, { "number": 72, "equation": "x² - 11x - 26 = 0", "solution": "a = 1, b = -11, c = -26.D = (-11)² - 4 * 1 * (-26) = 121 + 104 = 225.√D = √225 = 15.x₁ = (11 + 15) / (2 * 1) = 26 / 2 = 13.x₂ = (11 - 15) / (2 * 1) = -4 / 2 = -2.", "roots": "x₁ = 13, x₂ = -2" }, { "number": 73, "equation": "x² - 3x - 54 = 0", "solution": "a = 1, b = -3, c = -54.D = (-3)² - 4 * 1 * (-54) = 9 + 216 = 225.√D = √225 = 15.x₁ = (3 + 15) / (2 * 1) = 18 / 2 = 9.x₂ = (3 - 15) / (2 * 1) = -12 / 2 = -6.", "roots": "x₁ = 9, x₂ = -6" }, { "number": 74, "equation": "x² - 12x - 64 = 0", "solution": "a = 1, b = -12, c = -64.D = (-12)² - 4 * 1 * (-64) = 144 + 256 = 400.√D = √400 = 20.x₁ = (12 + 20) / (2 * 1) = 32 / 2 = 16.x₂ = (12 - 20) / (2 * 1) = -8 / 2 = -4.", "roots": "x₁ = 16, x₂ = -4" }, { "number": 75, "equation": "x² - 2x - 80 = 0", "solution": "a = 1, b = -2, c = -80.D = (-2)² - 4 * 1 * (-80) = 4 + 320 = 324.√D = √324 = 18.x₁ = (2 + 18) / (2 * 1) = 20 / 2 = 10.x₂ = (2 - 18) / (2 * 1) = -16 / 2 = -8.", "roots": "x₁ = 10, x₂ = -8" } ] }