Proof for an equality involving square roots

Question

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.

Answer 1

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 $$ (1+2a-a^2)\sqrt{x+k}=(1+2a-a^2)\sqrt{x-k} $$

$\ldots$