Answer
no solution
Work Step by Step
Square both sides to obtain:
$(\sqrt{x+2}+5)^2=(\sqrt{x+15})^2
\\(\sqrt{x+2}^2+2(\sqrt{x+2})(5)+5^2=x+15
\\x+2+10\sqrt{x+2}+25=x+15
\\x+10\sqrt{x+2}+27=x+15$
Isolate the terms with $x$ on the left side to obtain:
$x+10\sqrt{x+2}-x=15-27
\\10\sqrt{x+2}=-12$
Divide 10 to both sides:
$\dfrac{10\sqrt{x+2}}{10}=\dfrac{-12}{10}
\\\sqrt{x+2}=-\dfrac{6}{5}$
RECALL:
The square root of any real number is greater than or equal to zero.
Thus, the value of $\sqrt{x+2}$ is greater than or equal to zero.
This means that the value of $\sqrt{x+2}$ cannot be negative.
Therefore, the given equation has no solution.
