<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] Rationalise the denominator to get $$\frac{2c^2+4-2c\sqrt{c^2+4}}{2c^2+4}=\frac a{a-1}.$$ Then after simplifying, you get $$\frac 1{1-a}=\frac c{c^2+2}\sqrt{c^2+4}.$$ If you then square both sides and make the substitution $B=1/(1-a)^2$, you should get a quadratic equation in $c^2$, which ultimately leads where you're going.
Better still, use a computer algebra system.
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