Solve the following equation:

$$
\sqrt{x-2}+\sqrt{3-x}=x^{2}-5 x+7
$$
We seek the solution in the interval $2 \leq x \leq 3$, as the left side is only defined for these values. On the right side, $(x-2)(x-3)+1$ stands, and the product is negative within the given interval, and zero at the endpoints. We will show that the value of the left side is at least 1.

The square root of a non-negative number not greater than one is greater than or equal to the number itself. On the left side, we have the sum of the square roots of two such numbers, which is therefore not less than the sum of the numbers under the square roots. This sum is 1, so the value of the left side is indeed at least 1. Thus, equality can only hold if both sides have a value of 1, that is, at most when $x=2$ or $x=3$. By substitution, it can be seen that both are solutions to the equation.