Решите уравнение: (x-1)(x-2)(x-3)(x-4)=840.

Ответ:

\[(x - 1)(x - 2)(x - 3)(x + 4) =\]

\[= 840\]

\[\left( x^{2} - 5x + 4 \right)\left( x^{2} - 5x + 6 \right) =\]

\[= 840\]

\[Пусть\ x^{2} - 5x + 4 = y:\]

\[y(y + 2) = 840\]

\[y^{2} + 2y - 840 = 0\]

\[D = 1 + 840 = 841 = 29²\]

\[y_{1} = - 1 + 29 = 28;\ \ \ \]

\[y_{2} = - 1 - 29 = - 30.\]

\[Подставим:\]

\[1)\ x^{2} - 5x + 4 = 28\]

\[x^{2} - 5x - 24 = 0\]

\[x_{1} + x_{2} = 5;\ \ \ x_{1} \cdot x_{2} = - 24\]

\[x_{1} = 8;\ \ x_{2} = - 3.\]

\[2)\ x^{2} - 5x + 4 = - 30\]

\[x^{2} - 5x + 34 = 0\]

\[D = 25 - 4 \cdot 34 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 3;\ \ x = 8.\]