Answer
$x=20$
Work Step by Step
$\sqrt{\sqrt{x+5}+x}=5$ (Take the square)
$\sqrt{x+5}+x=25$
$\sqrt{x+5}=25-x$ (Take the square)
$x+5=x^2-50x+625$ (Simplify)
$x^2-51x+620=0$
$(x-20)(x-31)=0$
$x=20$ or $x=31$
Let us check these values of $x$ by back substituting to the original equation.
For $x=20$,
$\sqrt{\sqrt{20+5}+20}=5$
$\sqrt{5+20}=5$
$\sqrt{25}=5$ (True)
For $x=31$,
$\sqrt{\sqrt{31+5}+31}=5$
$\sqrt{\sqrt{36}+31}=5$
$\sqrt{6+31}=5$
$\sqrt{37}=5$ (False)
Therefore, the real solution is only $x=20$.
