Решите уравнение (x^2-x+1)(x^2-x-2)=378.

Ответ:

\[\left( x^{2} - x + 1 \right)\left( x^{2} - x - 2 \right) = 378\]

\[Пусть\ t = x^{2} - x + 1:\]

\[t(t - 3) = 378\]

\[t^{2} - 3t - 378 = 0\]

\[t_{1} + t_{2} = 3;\ \ \ t_{1} \cdot t_{2} = - 378\]

\[t_{1} = 21;\ \ \ \ t_{2} = - 18.\]

\[Подставим:\]

\[1)\ x^{2} - x + 1 = 21\]

\[x^{2} - x - 20 = 0\]

\[D = 1 + 80 = 81\]

\[x_{1} = \frac{1 - 9}{2} = - 4;\ \ \ x_{2} = \frac{1 + 9}{2} = 5.\]

\[2)\ x^{2} - x + 1 = - 18\]

\[x^{2} - x + 19 = 0\]

\[D = 1 - 76 = - 75 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 4;\ \ x = 5.\]