Решите уравнения из группы C: 1) $$(2x^2 + 7x - 8)(2x^2 + 7x - 3) = 314$$, 2) $$2y^3 + 2y^2 - (y + 1)^2 = 0$$, 3) $$6x^3 - 31x^2 - 31x + 6 = 0$$, 4) $$3x^2(x - 1)(x + 1) - 10x^2 + 4 = 0$$

**Решение уравнений группы C:** 1) $$(2x^2 + 7x - 8)(2x^2 + 7x - 3) = 314$$ * Замена: $$t = 2x^2 + 7x$$, получаем $$(t - 8)(t - 3) = 314$$ * $$t^2 - 11t + 24 = 314$$ * $$t^2 - 11t - 290 = 0$$ * $$D = (-11)^2 - 4 cdot 1 cdot (-290) = 121 + 1160 = 1281$$ * $$t_{1,2} = rac{11 pm sqrt{1281}}{2}$$ * Возвращаемся к замене: $$2x^2 + 7x = rac{11 pm sqrt{1281}}{2}$$ * $$4x^2 + 14x - (11 pm sqrt{1281}) = 0$$ * $$x = rac{-7 pm sqrt{49 + 4(11 pm sqrt{1281})}}{4} = rac{-7 pm sqrt{93 pm 4sqrt{1281}}}{4}$$. Это сложное выражение, но решение возможно. 2) $$2y^3 + 2y^2 - (y + 1)^2 = 0$$ * $$2y^3 + 2y^2 - (y^2 + 2y + 1) = 0$$ * $$2y^3 + y^2 - 2y - 1 = 0$$ * Сгруппируем: $$y^2(2y + 1) - (2y + 1) = 0$$ * $$(2y + 1)(y^2 - 1) = 0$$ * $$(2y + 1)(y - 1)(y + 1) = 0$$ * Ответ: $$y_1 = -rac{1}{2}, y_2 = 1, y_3 = -1$$ 3) $$6x^3 - 31x^2 - 31x + 6 = 0$$ * Уравнение симметричное. Разделим на $$x^3$$: $$6 - rac{31}{x} - rac{31}{x^2} + rac{6}{x^3}=0$$ * Сгруппируем: $$6(x^3 + rac{1}{x^3}) - 31(x + rac{1}{x}) = 0$$ * Замена: $$t = x + rac{1}{x}$$, тогда $$t^3 = x^3 + 3x + rac{3}{x} + rac{1}{x^3} = x^3 + rac{1}{x^3} + 3(x + rac{1}{x})$$, т.е. $$x^3 + rac{1}{x^3} = t^3 - 3t$$ * $$6(t^3 - 3t) - 31t = 0 Rightarrow 6t^3 - 18t - 31t = 0$$ * $$6t^3 - 49t = 0 Rightarrow t(6t^2 - 49) = 0$$ * $$t = 0$$ или $$6t^2 - 49 = 0$$ * 1) $$t = x + rac{1}{x} = 0 Rightarrow x^2 + 1 = 0$$. Корней нет. * 2) $$6t^2 = 49 Rightarrow t = pmsqrt{rac{49}{6}} = pmrac{7}{sqrt{6}}$$ * $$x + rac{1}{x} = rac{7}{sqrt{6}} Rightarrow sqrt{6}x^2 - 7x + sqrt{6} = 0$$. $$D = 49 - 4sqrt{6}sqrt{6} = 49 - 24 = 25$$. $$x_{1,2} = rac{7 pm 5}{2sqrt{6}} = rac{12}{2sqrt{6}} = sqrt{6}$$ or $$x=rac{2}{2sqrt{6}}=rac{1}{sqrt{6}}$$ * $$x + rac{1}{x} = -rac{7}{sqrt{6}} Rightarrow sqrt{6}x^2 + 7x + sqrt{6} = 0$$. $$D = 49 - 24 = 25$$. $$x_{3,4} = rac{-7 pm 5}{2sqrt{6}} = rac{-12}{2sqrt{6}} = -sqrt{6}$$ or $$x=rac{-2}{2sqrt{6}}=-rac{1}{sqrt{6}}$$ * Ответ: $$x_1 = sqrt{6}, x_2 = rac{1}{sqrt{6}}, x_3 = -sqrt{6}, x_4 = -rac{1}{sqrt{6}}$$ 4) $$3x^2(x - 1)(x + 1) - 10x^2 + 4 = 0$$ * $$3x^2(x^2 - 1) - 10x^2 + 4 = 0$$ * $$3x^4 - 3x^2 - 10x^2 + 4 = 0$$ * $$3x^4 - 13x^2 + 4 = 0$$ * Замена: $$t = x^2$$, получаем $$3t^2 - 13t + 4 = 0$$ * $$D = (-13)^2 - 4 cdot 3 cdot 4 = 169 - 48 = 121$$ * $$t_{1,2} = rac{13 pm sqrt{121}}{6} = rac{13 pm 11}{6}$$ * $$t_1 = rac{24}{6} = 4$$ * $$t_2 = rac{2}{6} = rac{1}{3}$$ * Возвращаемся к замене: $$x^2 = 4$$ или $$x^2 = rac{1}{3}$$ * $$x^2 = 4 Rightarrow x = pm 2$$ * $$x^2 = rac{1}{3} Rightarrow x = pm rac{1}{sqrt{3}} = pm rac{sqrt{3}}{3}$$ * Ответ: $$x_1 = 2, x_2 = -2, x_3 = rac{sqrt{3}}{3}, x_4 = -rac{sqrt{3}}{3}$$