Решите уравнение: (x-4)^4+(x-10)^4=272.

\[(x - 4)^{4} + (x - 10)^{4} = 272\]

\[t = \frac{x - 4 + x - 10}{2} = \frac{2x - 14}{2} =\]

\[= x - 7\]

\[(t + 3)^{4} + (t - 3)^{4} = 272\]

\[2t^{4} + 2 \cdot 54t^{2} - 110 = 0\]

\[2t^{4} + 108t^{2} - 110 = 0\ \ \ \ \ \ \ |\ :2\]

\[t^{4} + 54t^{2} - 55 = 0\]

\[t^{2} = y;\ \ \ y \geq 0\]

\[y^{2} + 54y - 55 = 0\]

\[D = 54^{2} - 4 \cdot 1 \cdot ( - 55) =\]

\[= 2916 + 220 = 3136\]

\[y_{1} = \frac{- 54 + \sqrt{3136}}{2} =\]

\[= \frac{- 54 + 56}{2} = \frac{2}{2} = 1\]

\[y_{2} = \frac{- 54 - \sqrt{3136}}{2} =\]

\[= \frac{- 54 - 56}{2} = \frac{- 110}{2} =\]

\[= - 55\ (не\ подходит).\]

\[Ответ:8;6.\]