Proof for an equality involving square roots While trying to solve this problem, I stumbled upon the following equality
$$
\sqrt{\sqrt{2x}+\sqrt{x+k}}+\sqrt{\sqrt{2x}+\sqrt{x-k}}=(\sqrt2+1)
\left(
\sqrt{\sqrt{2x}-\sqrt{x+k}}+\sqrt{\sqrt{2x}-\sqrt{x-k}}
\right)
$$
It seems to hold for any $x$ and $k$ in $\mathbb R$ such that all the square roots are well defined. If it can be proven, the proof of the above mentioned problem is immediate. However, I couldn't find a proof yet. I always have some surds left over. 

Denote $a=\sqrt{2}+1$ and write the equation as follows:
$$
\sqrt{\sqrt{2x}+\sqrt{x+k}}-a\sqrt{\sqrt{2x}-\sqrt{x+k}}=a\sqrt{\sqrt{2x}-\sqrt{x-k}}-\sqrt{\sqrt{2x}+\sqrt{x-k}}
$$
Squaring both sides, we get
$$
(1+2a-a^2)\sqrt{x+k}=(1+2a-a^2)\sqrt{x-k}
$$
$\ldots$
