<s>[INST] How to prove that $\sqrt{-3-2i}+\sqrt{-3+2i} = \sqrt{2(\sqrt{13}-3)}$? Is there a trick to show that
$$\sqrt{-3-2i}+\sqrt{-3+2i} = \sqrt{2(\sqrt{13}-3)}$$ 
is true ?
 [/INST] Consider the following.
\begin{align}
\sqrt{-3-2i}+\sqrt{-3+2i} &= \sqrt{a} \\
(\sqrt{-3 - 2 i} + \sqrt{-3 + 2 i} )^2 &= a \\
-6 + \sqrt{(-3-2i)(-3+2i)} &= a \\
(-3-2i)(-3+2i) &= (a+6)^2 \\
(a+6)^2 - 13 &=0 \\
a^2 + 12 a + 23 &= 0 \\
\end{align}
from here it is determined that $a = -6 \pm \sqrt{13}$, which yields the desired result.
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