We may be tempted to square the initial equation, but this would make the problem much more difficult. We need to move one of the terms to the other side prior to squaring both sides.
$$ \sqrt{x+22}-5\sqrt{x-2} = 0 $$
$$ \sqrt{x+22}=5\sqrt{x-2} $$
$$ (\sqrt{x+22})^2=(5\sqrt{x-2})^2 $$
$$ x+22=25(x-2) $$
$$ x+22=25x-50 $$
$$ 72=24x $$
$$ x=\boxed{3} $$
Substitute the options into the initial equation. Below is the work for the correct answer \(\boxed{3}\).
$$ \sqrt{x+22}-5\sqrt{x-2} = 0 $$
$$ \sqrt{(3)+22}-5\sqrt{(3)-2} = 0 $$
$$ \sqrt{25}-5\sqrt{1} = 0 $$
$$ 5-5(1)=0 $$
$$ 0=0 \checkmark $$