Решите уравнение: (x+1)(x+3)(x+5)(x+7)=945.

Ответ:

\[(x + 1)(x + 3)(x + 5)(x + 7) =\]

\[= 945\]

\[\left( x^{2} + 8x + 7 \right)\left( x^{2} + 8x + 15 \right) =\]

\[= 945\]

\[Пусть\ y = x^{2} + 8x + 7:\]

\[y(y + 8) = 945\]

\[y^{2} + 8y - 945 = 0\]

\[D = 16 + 945 = 961 = 31^{2}\]

\[y_{1} = - 4 + 31 = 27;\ \ \]

\[y_{2} = - 4 - 31 = - 35.\]

\[Подставим:\]

\[1)\ x^{2} + 8x + 7 = 27\]

\[x^{2} + 8x - 20 = 0\]

\[D = 16 + 20 = 36\]

\[x_{1} = - 4 + 6 = 2;\ \ \]

\[x_{2} = - 4 - 6 = - 10.\]

\[2)\ x^{2} + 8x + 7 = - 32\]

\[x^{2} + 8x + 39 = 0\]

\[D = 4 - 39 = - 35 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 10;\ \ x = 2.\]