finding solutions to a complex number equation

Question

Given that the square roots of $(-2+2\sqrt{3}\cdot{i})$ are $\pm(1+\sqrt{3}\cdot{i})$,

find all solutions to $\{z:z^2+(\sqrt{3}-i)z+(1-\sqrt{3}\cdot{i})=0\}$ in Cartesian form.

I'm unsure as to how to solve this question.

Answer 1

Solve as each quadratic equation:

$z^2+(\sqrt{3}-i)z+(1-\sqrt{3}\cdot{i})=0$

$D=(\sqrt{3}-i)^2-4(1-\sqrt{3}\cdot{i})=(-2+2\sqrt{3}\cdot{i})\Rightarrow z_{1,2}=\frac{-(\sqrt{3}-i)\pm \sqrt{-2+2\sqrt{3}\cdot{i}}}{2}=$

$=\frac{-(\sqrt{3}-i)\pm (1+\sqrt{3}\cdot{i})}{2}= \cdots$