Решите уравнение: (x-2)(x-3)(x-4)(x-5)=24.

\[(x - 2)(x - 3)(x - 4)(x - 5) =\]

\[= 24\]

\[(x - 2)(x - 5)(x - 3)(x - 4) =\]

\[= 24\]

\[\left( x^{2} - 7x + 10 \right)\left( x^{2} - 7x + 12 \right) =\]

\[= 24\]

\[t = x² - 7x + 10\]

\[t(t + 2) - 24 = 0\]

\[t^{2} + 2t - 24 = 0\]

\[D = 2^{2} - 4 \cdot 1 \cdot ( - 24) =\]

\[= 4 + 96 = 100\]

\[t_{1} = \frac{- 2 + \sqrt{100}}{2} = \frac{- 2 + 10}{2} =\]

\[= \frac{8}{2} = 4\]

\[t_{2} = \frac{- 2 - \sqrt{100}}{2} = \frac{- 2 - 10}{2} =\]

\[= \frac{- 12}{2} = - 6\]

\[1)\ x^{2} - 7x + 10 = 4\]

\[x^{2} - 7x + 6 = 0\]

\[D = ( - 7)^{2} - 4 \cdot 1 \cdot 6 =\]

\[= 49 - 24 = 25\]

\[x_{1} = \frac{7 + \sqrt{25}}{2} = \frac{7 + 5}{2} = \frac{12}{2} = 6\]

\[x_{2} = \frac{7 - \sqrt{25}}{2} = \frac{7 - 5}{2} = \frac{2}{2} = 1\]

\[2)\ x^{2} - 7x + 10 = - 6\]

\[x^{2} - 7x + 16 = 0\]

\[D = ( - 7) - 7 \cdot 1 \cdot 16 =\]

\[= 49 - 64 = - 15 < 0 \Longrightarrow\]

\[\Longrightarrow нет\ решения.\]

\[Ответ:6;1.\]