8.1. Find all real solutions of the equation $x^{3}-9(1-x)^{2}=x$.

Find all real solutions of the equation $x^{3}-9(1-x)^{2}=x$.
Solution: $\quad x^{3}-9(1-x)^{2}=x \Leftrightarrow x^{3}-x-9(x-1)^{2}=0 \quad \Leftrightarrow \quad x(x-1)(x+1)-9(x-1)^{2}=0 \Leftrightarrow$ $(x-1)\left(x^{2}-8 x+9\right)=0 \Leftrightarrow\left[\begin{array}{c}x-1=0 \\ x^{2}-8 x+9=0\end{array} \Leftrightarrow x \in\left\{1, \frac{8-2 \sqrt{7}}{2}, \frac{8+2 \sqrt{7}}{2}\right\} \Leftrightarrow x \in\{1,4-\sqrt{7}, 4+\sqrt{7}\}\right.$.

Answer: $S=\{1,4-\sqrt{7}, 4+\sqrt{7}\}$.