Answer
$3,7$
Work Step by Step
Step 1. Rewrite the equation as $\sqrt {2x-5}=1+\sqrt {x-3}$
Step 2. Take square on both sides to get $2x-5=1+2\sqrt {x-3}+x-3\longrightarrow 2\sqrt {x-3}=x-3$.
Thus $\sqrt {x-3}=0$ and $\sqrt {x-3}=2$
Step 3. We get the solutions $x=3$ and $x=2^2+3=7$
Step 4. Check the answers with the original equation, we find they both fit.