125. Решите уравнение: 1) x² - 6x - 27 = 0; 2) x² - 8x + 15 = 0; 3) 7y² - 4y - 3 = 0; 4) 6p² - p - 2 = 0; 5) x² + 4x - 10 = 0; 6) 4x² - 2x - 5 = 0; 7) 64x² - 48x + 9 = 0; 8) x² - 12x + 40 = 0.

Решение:

Для решения квадратных уравнений будем использовать формулу корней: \( x = \frac{-b \pm \sqrt{D}}{2a} \), где \( D = b^2 - 4ac \).

1. \( x^2 - 6x - 27 = 0 \)
   \( a = 1, b = -6, c = -27 \)
   \( D = (-6)^2 - 4 \cdot 1 \cdot (-27) = 36 + 108 = 144 \)
   \( x_{1,2} = \frac{6 \pm \sqrt{144}}{2 \cdot 1} = \frac{6 \pm 12}{2} \)
   \( x_1 = \frac{6 + 12}{2} = 9 \)
   \( x_2 = \frac{6 - 12}{2} = -3 \)
2. \( x^2 - 8x + 15 = 0 \)
   \( a = 1, b = -8, c = 15 \)
   \( D = (-8)^2 - 4 \cdot 1 \cdot 15 = 64 - 60 = 4 \)
   \( x_{1,2} = \frac{8 \pm \sqrt{4}}{2 \cdot 1} = \frac{8 \pm 2}{2} \)
   \( x_1 = \frac{8 + 2}{2} = 5 \)
   \( x_2 = \frac{8 - 2}{2} = 3 \)
3. \( 7y^2 - 4y - 3 = 0 \)
   \( a = 7, b = -4, c = -3 \)
   \( D = (-4)^2 - 4 \cdot 7 \cdot (-3) = 16 + 84 = 100 \)
   \( y_{1,2} = \frac{4 \pm \sqrt{100}}{2 \cdot 7} = \frac{4 \pm 10}{14} \)
   \( y_1 = \frac{4 + 10}{14} = 1 \)
   \( y_2 = \frac{4 - 10}{14} = -\frac{6}{14} = -\frac{3}{7} \)
4. \( 6p^2 - p - 2 = 0 \)
   \( a = 6, b = -1, c = -2 \)
   \( D = (-1)^2 - 4 \cdot 6 \cdot (-2) = 1 + 48 = 49 \)
   \( p_{1,2} = \frac{1 \pm \sqrt{49}}{2 \cdot 6} = \frac{1 \pm 7}{12} \)
   \( p_1 = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3} \)
   \( p_2 = \frac{1 - 7}{12} = -\frac{6}{12} = -\frac{1}{2} \)
5. \( x^2 + 4x - 10 = 0 \)
   \( a = 1, b = 4, c = -10 \)
   \( D = 4^2 - 4 \cdot 1 \cdot (-10) = 16 + 40 = 56 \)
   \( x_{1,2} = \frac{-4 \pm \sqrt{56}}{2 \cdot 1} = \frac{-4 \pm 2\sqrt{14}}{2} \)
   \( x_1 = -2 + \sqrt{14} \)
   \( x_2 = -2 - \sqrt{14} \)
6. \( 4x^2 - 2x - 5 = 0 \)
   \( a = 4, b = -2, c = -5 \)
   \( D = (-2)^2 - 4 \cdot 4 \cdot (-5) = 4 + 80 = 84 \)
   \( x_{1,2} = \frac{2 \pm \sqrt{84}}{2 \cdot 4} = \frac{2 \pm 2\sqrt{21}}{8} \)
   \( x_1 = \frac{1 + \sqrt{21}}{4} \)
   \( x_2 = \frac{1 - \sqrt{21}}{4} \)
7. \( 64x^2 - 48x + 9 = 0 \)
   \( a = 64, b = -48, c = 9 \)
   \( D = (-48)^2 - 4 \cdot 64 \cdot 9 = 2304 - 2304 = 0 \)
   \( x = \frac{48 \pm \sqrt{0}}{2 \cdot 64} = \frac{48}{128} = \frac{3}{8} \)
8. \( x^2 - 12x + 40 = 0 \)
   \( a = 1, b = -12, c = 40 \)
   \( D = (-12)^2 - 4 \cdot 1 \cdot 40 = 144 - 160 = -16 \)
   \( D < 0 \), действительных корней нет.

Ответ: 1) 9; -3; 2) 5; 3; 3) 1; -3/7; 4) 2/3; -1/2; 5) -2 + √14; -2 - √14; 6) (1 + √21)/4; (1 - √21)/4; 7) 3/8; 8) действительных корней нет.