Решите уравнение: (x-1)(x-3)(x-5)(x-7)=105.

Ответ:

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

\[= 105\]

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

\[= 105\]

\[Пусть\ \left( x^{2} - 8x + 7 \right) = a:\]

\[a(a + 8) = 105\]

\[a^{2} + 8a - 105 = 0\]

\[D = 16 + 105 = 121\]

\[a_{1} = - 4 + 11 = 7;\ \ \ \]

\[a_{2} = - 4 - 11 = - 15.\]

\[Подставим:\]

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

\[x^{2} - 8x + 7 - 7 = 0\]

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

\[x(x - 8) = 0\]

\[x = 0;\ \ \ x = 8.\]

\[2)\ x^{2} - 8x + 7 = - 15\]

\[x^{2} - 8x + 22 = 0\]

\[D = 16 - 22 < 0\]

\[нет\ корней.\]

\[Ответ:\ \ x = 0;\ \ x = 8.\]