Answer
$x=20$
Work Step by Step
$\sqrt{\sqrt{x+5}+x}=5$
Square both sides:
$(\sqrt{\sqrt{x+5}+x})^{2}=5^{2}$
$\sqrt{x+5}+x=25$
Take $x$ to the right side:
$\sqrt{x+5}=25-x$
Again, square both sides:
$(\sqrt{x+5})^{2}=(25-x)^{2}$
$x+5=625-50x+x^{2}$
Take all terms to the right side:
$0=x^{2}-50x-x+625-5$
$0=x^{2}-51x+620$
Reorganize:
$x^{2}-51x+620=0$
Solve this equation by factoring:
$(x-20)(x-31)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x-20=0$
$x=20$
$x-31=0$
$x=31$
Check your answer by plugging them into the original equation:
$x=20$
$\sqrt{\sqrt{20+5}+20}=5$
$\sqrt{\sqrt{25}+20}=5$
$\sqrt{5+20}=5$
$\sqrt{25}=5$
$5=5$ True
$x=31$
$\sqrt{\sqrt{31+5}+31}=5$
$\sqrt{\sqrt{36}+31}=5$
$\sqrt{6+31}=5$
$\sqrt{36}=5$
$6\ne5$ False
The final answer is $x=20$
