<s>[INST] solving $a = \sqrt{b + x} + \sqrt{c + x}$ for $x$ I'm trying to solve a very simple looking square root equation but nothing seems to work. The equation has this form (solve for $x$):
$$
a = \sqrt{b + x} + \sqrt{c + x}
$$
Squaring both sides obviously doesn't help since it will still give me a square root. Rearranging and then squaring doesn't help either.
The problem looks very simple to me but I have no idea on how to approach this.
 [/INST] $$\begin{align}a=\sqrt{b+x}+\sqrt{c+x}&\Rightarrow \sqrt{b+x}=a-\sqrt{c+x}\\&\Rightarrow b+x=a^2-2a\sqrt{c+x}+c+x\\&\Rightarrow 2a\sqrt{c+x}=a^2+c-b\\&\Rightarrow 4a^2(c+x)=(a^2+c-b)^2\\&\Rightarrow x=\frac{(a^2+c-b)^2-4a^2c}{4a^2}\end{align}$$
</s>