Problem 3. (15 points) Solve the equation $6 x-x^{2}+\sqrt{\left(25-x^{2}\right)\left(-11+12 x-x^{2}\right)}=11$.

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# Solution.

Rewrite the equation as $\sqrt{\left(25-x^{2}\right)\left(-11+12 x-x^{2}\right)}=x^{2}-6 x+11$,

$$
\sqrt{(x-5)(x+5)(x-1)(x-11)}=x^{2}-6 x+11 .
$$

Rearrange the factors $(x-5)(x-1)$ and $(x+5)(x-11)$, we get

$$
\sqrt{\left(x^{2}-6 x+5\right)\left(x^{2}-6 x-55\right)}=x^{2}-6 x+11
$$

Let $x^{2}-6 x+5=t$, then $x^{2}-6 x-55=t-60, x^{2}-6 x+11=t+6$, we get

$$
\begin{gathered}
\sqrt{t(t-40)}=t+5 \\
\left\{\begin{array}{l}
t+6 \geq 0 \\
t(t-60)=(t+6)^{2}
\end{array}\right. \\
t=-\frac{1}{2} \\
x^{2}-6 x+5=-\frac{1}{2} \\
2 x^{2}-12 x+11=0 \\
x_{1,2}=\frac{6 \pm \sqrt{14}}{2}
\end{gathered}
$$

Answer. $\left\{\frac{6 \pm \sqrt{14}}{2}\right\}$.