123. Решите уравнение: 1) x²+5x-14=0; 2) x²-14x+40=0; 3) 3y²-13y+4=0; 4) 12m²+m-6=0; 5) x²+6x-2=0; 6) 3x²-4x-5=0; 7) 25x²+60x+36=0; 8) x²-8x+18=0.

Привет! Разберемся с решением квадратных уравнений. 1) x² + 5x - 14 = 0; D = b² - 4ac = 5² - 4 * 1 * (-14) = 25 + 56 = 81 x₁ = (-b + √D) / 2a = (-5 + √81) / 2 = (-5 + 9) / 2 = 4 / 2 = 2 x₂ = (-b - √D) / 2a = (-5 - √81) / 2 = (-5 - 9) / 2 = -14 / 2 = -7

Ответ: x₁ = 2, x₂ = -7

2) x² - 14x + 40 = 0; D = (-14)² - 4 * 1 * 40 = 196 - 160 = 36 x₁ = (14 + √36) / 2 = (14 + 6) / 2 = 20 / 2 = 10 x₂ = (14 - √36) / 2 = (14 - 6) / 2 = 8 / 2 = 4

Ответ: x₁ = 10, x₂ = 4

3) 3y² - 13y + 4 = 0; D = (-13)² - 4 * 3 * 4 = 169 - 48 = 121 y₁ = (13 + √121) / 6 = (13 + 11) / 6 = 24 / 6 = 4 y₂ = (13 - √121) / 6 = (13 - 11) / 6 = 2 / 6 = 1/3

Ответ: y₁ = 4, y₂ = 1/3

4) 12m² + m - 6 = 0; D = 1² - 4 * 12 * (-6) = 1 + 288 = 289 m₁ = (-1 + √289) / 24 = (-1 + 17) / 24 = 16 / 24 = 2/3 m₂ = (-1 - √289) / 24 = (-1 - 17) / 24 = -18 / 24 = -3/4

Ответ: m₁ = 2/3, m₂ = -3/4

5) x² + 6x - 2 = 0; D = 6² - 4 * 1 * (-2) = 36 + 8 = 44 x₁ = (-6 + √44) / 2 = (-6 + 2√11) / 2 = -3 + √11 x₂ = (-6 - √44) / 2 = (-6 - 2√11) / 2 = -3 - √11

Ответ: x₁ = -3 + √11, x₂ = -3 - √11

6) 3x² - 4x - 5 = 0; D = (-4)² - 4 * 3 * (-5) = 16 + 60 = 76 x₁ = (4 + √76) / 6 = (4 + 2√19) / 6 = (2 + √19) / 3 x₂ = (4 - √76) / 6 = (4 - 2√19) / 6 = (2 - √19) / 3

Ответ: x₁ = (2 + √19) / 3, x₂ = (2 - √19) / 3

7) 25x² + 60x + 36 = 0; D = 60² - 4 * 25 * 36 = 3600 - 3600 = 0 x = -60 / (2 * 25) = -60 / 50 = -6/5 = -1.2

Ответ: x = -1.2

8) x² - 8x + 18 = 0. D = (-8)² - 4 * 1 * 18 = 64 - 72 = -8

Ответ: Нет действительных корней