<s>[INST] Equation with complex numbers and fractions I want to solve this, but I do not know how to continue after solving the square in the denominator. I tried multiplying by denominators, but I got awkward results.
I appreciate your help.

 [/INST] To simplify this I did the following. First expand out the square in the denominator.
$(1+i)^2=1+2i+i^2=2i\Rightarrow$
$z_2=\frac{-\frac{1}{\sqrt{3}}+2i}{5i+2i} + \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}-2i}$
Next multiply the second fractions numerator and denominator by the denominators conjugate.
$(\sqrt{3}-2i)(\sqrt{3}+2i)=3+4=7 \Rightarrow$
$z_2=\frac{-\frac{1}{\sqrt{3}}+2i}{7i} + \frac{\frac{\sqrt{3}+2i}{\sqrt{3}}}{7}=\frac{-\frac{1}{\sqrt{3}}+2i}{7i} + \frac{\sqrt{3}+2i}{7\sqrt{3}}$
Next we combine the fractions by multiplying them by the each others denominators.
$z_2=\frac{-7+14\sqrt{3}i+7\sqrt{3}i-14}{49\sqrt{3}i}=\frac{21\sqrt{3}i-21}{49\sqrt{3}i}=\frac{3\sqrt{3}i-3}{7\sqrt{3}i}$
Finally we simplify to get the answer.
$z_2=\frac{3}{7}-\frac{3}{7\sqrt{3}i}$
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