4. (10 points) Solve the equation

$$
\sqrt{7-x^{2}+6 x}+\sqrt{6 x-x^{2}}=7+\sqrt{x(3-x)}
$$
# Solution:

The domain of the variable x in our problem is the interval [0;3]. By completing the square in the expressions under the square roots on the left side of the equation or by plotting the graphs, we notice that the values of the first root do not exceed 4, and the second does not exceed 3, with the minimum values being reached at $\mathrm{x}=3$. The value of the left side of the equation does not exceed 7, while the right side is no less than 7. Therefore, equality is possible only when both sides of the equation are simultaneously equal to 7, which is achieved at $x=3$.

Answer: 3