Let $a$ and $b$ be distinct positive numbers. Solve the equation

$$
\sqrt{x-b^{2}}-\sqrt{x-a^{2}}=a-b
$$
Solution. We need to have $x \geq a^{2}, x \geq b^{2}$

The equation can be equivalently written as

$$
\frac{a^{2}-b^{2}}{\sqrt{x-b^{2}}+\sqrt{x-a^{2}}}=a-b
$$

from which we get

$$
\sqrt{x-b^{2}}+\sqrt{x-a^{2}}=a+b
$$

Adding this to the initial equation, we obtain

$$
\sqrt{x-b^{2}}=a
$$

$\qquad$
from which $x=a^{2}+b^{2}$, which is clearly a solution.