1. Найдите корни уравнения: 1) a) (x-2)² - 3x - 8; б) (x-1)² = 29 - 5x; в) 5(x+2)² = -6x - 44; г) (x+3)² - 16 = (1 - 2x)²; 2) a) (x-2)(x+2) = 7x - 14; б) (-x-1)(x-4) = x(4x-11); в) -x(1/3 - x) = (x-1)(x+1); г) 5(x-2) = (3x+2)(x-2); 3) a) (x²-x)/3 = (2x-4)/5; б) (x²-3)/2 - 6x = 5; в) (x²+2x)/2 = (x²+24)/7; г) (3x²+x)/4 - (2-7x)/5 = (3x²+17)/10.

Решение: 1) a) (x-2)² - 3x - 8 = 0 x² - 4x + 4 - 3x - 8 = 0 x² - 7x - 4 = 0 D = (-7)² - 4 * 1 * (-4) = 49 + 16 = 65 x₁ = (7 + √65) / 2 x₂ = (7 - √65) / 2 б) (x-1)² = 29 - 5x x² - 2x + 1 = 29 - 5x x² + 3x - 28 = 0 D = 3² - 4 * 1 * (-28) = 9 + 112 = 121 x₁ = (-3 + √121) / 2 = (-3 + 11) / 2 = 4 x₂ = (-3 - √121) / 2 = (-3 - 11) / 2 = -7 в) 5(x+2)² = -6x - 44 5(x² + 4x + 4) = -6x - 44 5x² + 20x + 20 = -6x - 44 5x² + 26x + 64 = 0 D = 26² - 4 * 5 * 64 = 676 - 1280 = -604 < 0 Нет действительных корней. г) (x+3)² - 16 = (1 - 2x)² x² + 6x + 9 - 16 = 1 - 4x + 4x² -3x² + 10x - 8 = 0 3x² - 10x + 8 = 0 D = (-10)² - 4 * 3 * 8 = 100 - 96 = 4 x₁ = (10 + √4) / 6 = (10 + 2) / 6 = 2 x₂ = (10 - √4) / 6 = (10 - 2) / 6 = 4/3 2) a) (x-2)(x+2) = 7x - 14 x² - 4 = 7x - 14 x² - 7x + 10 = 0 D = (-7)² - 4 * 1 * 10 = 49 - 40 = 9 x₁ = (7 + √9) / 2 = (7 + 3) / 2 = 5 x₂ = (7 - √9) / 2 = (7 - 3) / 2 = 2 б) (-x-1)(x-4) = x(4x-11) -x² + 4x - x + 4 = 4x² - 11x 5x² - 14x - 4 = 0 D = (-14)² - 4 * 5 * (-4) = 196 + 80 = 276 x₁ = (14 + √276) / 10 = (14 + 2√69) / 10 = (7 + √69) / 5 x₂ = (14 - √276) / 10 = (7 - √69) / 5 в) -x(1/3 - x) = (x-1)(x+1) -x/3 + x² = x² - 1 -x/3 = -1 x = 3 г) 5(x-2) = (3x+2)(x-2) 5x - 10 = 3x² - 6x + 2x - 4 3x² - 9x + 6 = 0 x² - 3x + 2 = 0 D = (-3)² - 4 * 1 * 2 = 9 - 8 = 1 x₁ = (3 + √1) / 2 = (3 + 1) / 2 = 2 x₂ = (3 - √1) / 2 = (3 - 1) / 2 = 1 3) a) (x²-x)/3 = (2x-4)/5 5(x²-x) = 3(2x-4) 5x² - 5x = 6x - 12 5x² - 11x + 12 = 0 D = (-11)² - 4 * 5 * 12 = 121 - 240 = -119 < 0 Нет действительных корней. б) (x²-3)/2 - 6x = 5 x² - 3 - 12x = 10 x² - 12x - 13 = 0 D = (-12)² - 4 * 1 * (-13) = 144 + 52 = 196 x₁ = (12 + √196) / 2 = (12 + 14) / 2 = 13 x₂ = (12 - √196) / 2 = (12 - 14) / 2 = -1 в) (x²+2x)/2 = (x²+24)/7 7(x² + 2x) = 2(x² + 24) 7x² + 14x = 2x² + 48 5x² + 14x - 48 = 0 D = 14² - 4 * 5 * (-48) = 196 + 960 = 1156 x₁ = (-14 + √1156) / 10 = (-14 + 34) / 10 = 2 x₂ = (-14 - √1156) / 10 = (-14 - 34) / 10 = -4.8 г) (3x²+x)/4 - (2-7x)/5 = (3x²+17)/10 5(3x² + x) - 4(2 - 7x) = 2(3x² + 17) 15x² + 5x - 8 + 28x = 6x² + 34 9x² + 33x - 42 = 0 3x² + 11x - 14 = 0 D = 11² - 4 * 3 * (-14) = 121 + 168 = 289 x₁ = (-11 + √289) / 6 = (-11 + 17) / 6 = 1 x₂ = (-11 - √289) / 6 = (-11 - 17) / 6 = -14/3