Задание 3. Решите уравнение: 1) x^6=(6x-5)^3; 2) x^6=(7x-12)^3; 3) x^6=(6x-8)^3; 4) x^3=3x^2+4x; 5) x^3=2x^2+8x; 6) x^3=4x^2+5x; 7) (x-2)^2(x-3)=20(x-2); 8) (x-2)^2(x-3)=12(x-2).

1) $x^6 = (6x - 5)^3$ $(x^2)^3 = (6x - 5)^3$ $x^2 = 6x - 5$ $x^2 - 6x + 5 = 0$ $D = (-6)^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16 = 4^2$ $x_1 = \frac{6 + 4}{2} = 5, \quad x_2 = \frac{6 - 4}{2} = 1$ **Ответ: 1; 5.** 2) $x^6 = (7x - 12)^3$ $(x^2)^3 = (7x - 12)^3$ $x^2 = 7x - 12$ $x^2 - 7x + 12 = 0$ $D = (-7)^2 - 4 \cdot 1 \cdot 12 = 49 - 48 = 1 = 1^2$ $x_1 = \frac{7 + 1}{2} = 4, \quad x_2 = \frac{7 - 1}{2} = 3$ **Ответ: 3; 4.** 3) $x^6 = (6x - 8)^3$ $(x^2)^3 = (6x - 8)^3$ $x^2 = 6x - 8$ $x^2 - 6x + 8 = 0$ $D = (-6)^2 - 4 \cdot 1 \cdot 8 = 36 - 32 = 4 = 2^2$ $x_1 = \frac{6 + 2}{2} = 4, \quad x_2 = \frac{6 - 2}{2} = 2$ **Ответ: 2; 4.** 4) $x^3 = 3x^2 + 4x$ $x^3 - 3x^2 - 4x = 0$ $x(x^2 - 3x - 4) = 0$ $x_1 = 0$ или $x^2 - 3x - 4 = 0$ $D = (-3)^2 - 4 \cdot 1 \cdot (-4) = 9 + 16 = 25 = 5^2$ $x_2 = \frac{3 + 5}{2} = 4, \quad x_3 = \frac{3 - 5}{2} = -1$ **Ответ: -1; 0; 4.** 5) $x^3 = 2x^2 + 8x$ $x^3 - 2x^2 - 8x = 0$ $x(x^2 - 2x - 8) = 0$ $x_1 = 0$ или $x^2 - 2x - 8 = 0$ $D = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36 = 6^2$ $x_2 = \frac{2 + 6}{2} = 4, \quad x_3 = \frac{2 - 6}{2} = -2$ **Ответ: -2; 0; 4.** 6) $x^3 = 4x^2 + 5x$ $x^3 - 4x^2 - 5x = 0$ $x(x^2 - 4x - 5) = 0$ $x_1 = 0$ или $x^2 - 4x - 5 = 0$ $D = (-4)^2 - 4 \cdot 1 \cdot (-5) = 16 + 20 = 36 = 6^2$ $x_2 = \frac{4 + 6}{2} = 5, \quad x_3 = \frac{4 - 6}{2} = -1$ **Ответ: -1; 0; 5.** 7) $(x - 2)^2(x - 3) = 20(x - 2)$ $(x - 2)^2(x - 3) - 20(x - 2) = 0$ $(x - 2)((x - 2)(x - 3) - 20) = 0$ $x_1 = 2$ или $x^2 - 5x + 6 - 20 = 0 \Rightarrow x^2 - 5x - 14 = 0$ $D = (-5)^2 - 4 \cdot 1 \cdot (-14) = 25 + 56 = 81 = 9^2$ $x_2 = \frac{5 + 9}{2} = 7, \quad x_3 = \frac{5 - 9}{2} = -2$ **Ответ: -2; 2; 7.** 8) $(x - 2)^2(x - 3) = 12(x - 2)$ $(x - 2)^2(x - 3) - 12(x - 2) = 0$ $(x - 2)((x - 2)(x - 3) - 12) = 0$ $x_1 = 2$ или $x^2 - 5x + 6 - 12 = 0 \Rightarrow x^2 - 5x - 6 = 0$ $D = (-5)^2 - 4 \cdot 1 \cdot (-6) = 25 + 24 = 49 = 7^2$ $x_2 = \frac{5 + 7}{2} = 6, \quad x_3 = \frac{5 - 7}{2} = -1$ **Ответ: -1; 2; 6.**