Task 3. (15 points) Solve the equation $4 x-x^{2}+\sqrt{\left(9-x^{2}\right)\left(-7+8 x-x^{2}\right)}=7$.

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Solution.

Rewrite the equation as $\sqrt{\left(9-x^{2}\right)\left(-7+8 x-x^{2}\right)}=x^{2}-4 x+7$.

$$
\sqrt{(x-3)(x+3)(x-1)(x-7)}=x^{2}-4 x+7
$$

Rearrange the factors $(x-3)(x-1)$ and $(x+3)(x-7)$, we get

$$
\sqrt{\left(x^{2}-4 x+3\right)\left(x^{2}-4 x-21\right)}=x^{2}-4 x+7
$$

Let $x^{2}-4 x+3=t$, then $x^{2}-4 x-21=t-24, x^{2}-4 x+7=t+4$, we get

$$
\begin{gathered}
\sqrt{t(t-24)}=t+4 \\
\left\{\begin{array}{l}
t+4 \geq 0 \\
t(t-24)=(t+4)^{2}
\end{array}\right. \\
t=-\frac{1}{2} \\
x^{2}-4 x+3=-\frac{1}{2} \\
2 x^{2}-8 x+7=0 \\
x_{1,2}=\frac{4 \pm \sqrt{2}}{2}
\end{gathered}
$$

Answer. $\frac{4 \pm \sqrt{2}}{2}$.