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.

HINT
If we agree that $\alpha = \sqrt{b + x}$ and $\beta = \sqrt{c + x}$ we obtain that $\alpha + \beta = a$ and $\alpha^{2} -\beta^{2} = b - c$. Hence it can be claimed $(\alpha + \beta)(\alpha - \beta) = a(\alpha - \beta) = b-c$ which implies the following system of equations
\begin{align*}
\begin{cases}
\alpha - \beta = \displaystyle\frac{b-c}{a}\\
\alpha + \beta = a
\end{cases} \Rightarrow 2\alpha  = a + \frac{b-c}{a} = \frac{a^{2} + b - c}{a}\Leftrightarrow \boxed{\alpha = \sqrt{b+x} = \frac{a^{2} + b - c}{2a}}
\end{align*}
