Решите уравнение, используя введение новой переменной: (x^2-x+1)(x^2-x-7)=65.

\[\left( x^{2} - x + 1 \right)\left( x^{2} - x - 7 \right) = 65\]

\[Пусть\ \left( x^{2} - x \right) = a:\]

\[(a + 1)(a - 7) = 65\]

\[a^{2} + a - 7a - 7 - 65 = 0\]

\[a^{2} - 6a - 72 = 0\]

\[D = 9 + 72 = 81\]

\[a_{1} = 3 + 9 = 12;\ \ \ \]

\[a_{2} = 3 - 9 = - 6.\]

\[Подставим:\]

\[1)\ x^{2} - x = 12\]

\[x^{2} - x - 12 = 0\]

\[x_{1} + x_{2} = 1;\ \ \ \ x_{1} \cdot x_{2} = - 12\]

\[x_{1} = 4;\ \ x_{2} = - 3.\]

\[2)\ x^{2} - x = - 6\]

\[x^{2} - x + 6 = 0\]

\[D = 1 - 24 < 0 - нет\ корней.\]

\[Ответ:\ \ x = - 3;\ \ x = 4.\]