Решите уравнение: ||x^2-4x+1|-1|=2.

\[\left| \left| x^{2} - 4x + 1 \right| - 1 \right| = 2\]

\[\left| x^{2} - 4x + 1 \right| - 1 = 2\]

\[\left| x^{2} - 4x + 1 \right| = 3\]

\[x^{2} - 4x + 1 = 3\]

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

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

\[= 16 + 8 = 24\]

\[x_{1} = \frac{4 + \sqrt{24}}{2} = \frac{4 + 2\sqrt{6}}{2} =\]

\[= 2 + \sqrt{6}\]

\[x_{2} = \frac{4 - \sqrt{24}}{2} = \frac{4 - 2\sqrt{6}}{2} =\]

\[= 2 - \sqrt{6}\]

\[x^{2} - 4x + 1 = - 3\]

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

\[(x - 2)^{2} = 0\]

\[x - 2 = 0\]

\[x = 2.\]

\[\left| x^{2} - 4x + 1 \right| - 1 = - 2\]

\[\left| x^{2} - 4x + 1 \right| = - 1 \Longrightarrow\]

\[\Longrightarrow нет\ решения.\]

\[Ответ:2 + \sqrt{6}\ ;\ 2 - \sqrt{6};\ \ 2.\]