Найдите корни уравнения (x-2)(x-1)(x+2)(x+3)=60.

Ответ:

\[(x - 2)(x - 1)(x + 2)(x + 3) = 60\]

\[\left( (x - 2)(x + 3) \right)\left( (x - 1)(x + 2) \right) = 60\]

\[\left( x^{2} + x - 6 \right)\left( x^{2} + x - 2 \right) = 60\]

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

\[y(y + 4) = 60\]

\[y^{2} + 4y - 60 = 0\]

\[D = 4 + 60 = 64\]

\[y_{1} = - 2 + 8 = 6;\ \ y_{2} = - 2 - 8 = - 10\]

\[1)\ y = - 10:\]

\[x^{2} + x - 6 = - 10\]

\[x^{2} + x + 4 = 0\]

\[D = 1 - 16 < 0\]

\[нет\ корней.\]

\[2)\ y = 6:\]

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

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

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

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

\[Ответ:x = - 4;\ \ x = 3.\]