<s>[INST] How to solve this equation for c? I want to solve the equation
$$
-\frac{a}{2}\left(c+\sqrt{c^2+4}\right)=-\frac{a-1}{2}\left(c-\sqrt{c^2+4}\right)
$$
for $c$, where $a$ is just a constant.
What I get is
$$
\frac{c-\sqrt{c^2+4}}{c+\sqrt{c^2+4}}=\frac{a}{a-1}.
$$
I think there now is some "trick" to solve this for $c$.
 [/INST] Try to isolate the square root, it is much simpler:
$$-a \sqrt{c^2+4}+a c+\sqrt{c^2+4}-c=a \sqrt{c^2+4}+a c\\
-2 a \sqrt{c^2+4}+\sqrt{c^2+4}-c=0\\
(1-2 a) \sqrt{c^2+4}-c=0\\
 \sqrt{c^2+4}=\frac{c}{1-2a}$$
Square both sides and you get:
$$c^2+4=\frac{c^2}{(1-2a)^2}$$
Rearrange that and you have a quadratic equation:
$$((1-2a)^2-1)c^2+4(1-2a)^2=0$$
Using the quadratic formula gives you:
$$c=\begin{cases}
-\frac{\sqrt{-4 a^2+4 a-1}}{\sqrt{a^2-a}}\Rightarrow -\frac{i (1-2 a)}{\sqrt{a-1} \sqrt{a}}\\
\frac{\sqrt{-4 a^2+4 a-1}}{\sqrt{a^2-a}}\Rightarrow \frac{i (1-2 a)}{\sqrt{a-1} \sqrt{a}}
\end{cases}$$
</s>