3. [4 points] Solve the equation $\frac{1}{2}(x+5) \sqrt{x^{3}-16 x+25}=x^{2}+3 x-10$.
Answer: $3 ; \frac{\sqrt{13}+1}{2}$.

Solution. Factoring the right side, we get $(x+5) \sqrt{x^{3}-16 x+25}=(x+$ 5) $(2 x-4)$. From here, there are two possibilities: either $x+5=0$ (then $x=-5$, which does not fit the domain of definition, as the expression under the root is negative), or $\sqrt{x^{3}-16 x+25}=2 x-4$. We solve the latter equation:

$$
\left\{\begin{array} { l } 
{ x ^ { 3 } - 1 6 x + 2 5 = ( 2 x - 4 ) ^ { 2 } , } \\
{ 2 x - 4 \geqslant 0 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ x ^ { 3 } - 4 x ^ { 2 } + 9 = 0 , } \\
{ x \geqslant 2 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
(x-3)\left(x^{2}-x-3\right)=0, \\
x \geqslant 2 .
\end{array}\right.\right.\right.
$$

The equation of the system has roots $x=3, x=\frac{1 \pm \sqrt{13}}{2}$, and of these, the inequality is satisfied by $x=3$ and $x=\frac{1+\sqrt{13}}{2}$. These are the answers to the problem.