Answer
Solution set: $\{18\}$
Work Step by Step
$\sqrt{3x-5}-\sqrt{x+7}=2$
...Add $\sqrt{x+7}$ to both sides
$\sqrt{3x-5}=2+\sqrt{x+7} ...$ Square both sides
$(\sqrt{3x-5})^{2}=(2+\sqrt{x+7})^{2}$
...Apply $(a+b)^{2}=a^{2}+2ab+b^{2}$
$3x-5=4+4\sqrt{x+7}+(x+7)$
$3x-5=x+4\sqrt{x+7}+11$
$2x-16=4\sqrt{x+7} ...$ Square both sides
$4x^{2}-64x+256=16(x+7)$
$4x^{2}-64x+256=16x+112$
$4x^{2}-80x+144=0$
$x^{2}-20x+36=0$
$(x-18)(x-2)=0$
Testing $x=18$
$\sqrt{3(18)-5}-\sqrt{18+7}=7-5=2$
... $18$ is a solution.
Testing $x=2$
$\sqrt{3(2)-5}-\sqrt{2+7}=1-3\neq 2$
... $2$ is not a solution.
Solution set: $\{18\}$
