Problem source

\begin{questionparts}
\item Find all rational numbers $r$ and $s$ which satisfy 
\[
(r+s\sqrt{3})^{2}=4-2\sqrt{3}.
\]
\item Find all real numbers $p$ and $q$ which satisfy 
\[
(p+q\mathrm{i})^{2}=(3-2\sqrt{3})+2(1-\sqrt{3})\mathrm{i}.
\]
\item Solve the equation 
\[
(1+\mathrm{i})z^{2}-2z+2\sqrt{3}-2=0,
\]
writing your solutions in as simple a form as possible. 
\end{questionparts}
{[}No credit will be given to answers involving use of calculators.{]}

Solution source

\begin{questionparts}
\item Suppose 
\begin{align*}
&& 4 - 2\sqrt{3} &= (r+s\sqrt{3})^2 \\
&&&= r^2+3s^2+2sr \sqrt{3} \\
\Rightarrow && rs &= -1 \\
&& r^2+3s^2 &= 4 \\
\Rightarrow && (r,s) &= (1,-1), (-1,1)
\end{align*}
\item \begin{align*}
&&  (3-2\sqrt{3})+2(1-\sqrt{3})i  &= (p+qi)^2 \\
&&&= p^2-q^2 + 2pq i \\
\Rightarrow && pq &= (1-\sqrt{3}) \\
&& p^2 - q^2 &= 3-2\sqrt{3} \\
\Rightarrow &&3-2\sqrt{3} &= p^2 - \frac{(1-\sqrt{3})^2}{p^2} \\
\Rightarrow && 0 &= p^4-(3-2\sqrt{3})p^2-(4-2\sqrt{3}) \\
&&&= (p^2-(4-2\sqrt{3}))(p^2+1) \\
\Rightarrow && p &= \pm (1-\sqrt{3}) \\
&& q &=\mp \frac12(1+\sqrt{3})
\end{align*}
\item \begin{align*}
&& 0 &= (1+i)z^2 - 2z + 2(\sqrt{3}-1) \\
\Rightarrow && z &= \frac{2 \pm \sqrt{4-4(1+i)2(\sqrt{3}-1)}}{2(1+i)} \\
&&&= \frac{1 \pm \sqrt{1-(1+i)2(\sqrt{3}-1)}}{1+i} \\
&&&= \frac{1 \pm \sqrt{(3-2\sqrt{3})+(2-2\sqrt{3})i}}{1+i} \\
&&&= \frac{1 \pm (1 - \sqrt{3}) \mp \frac12 (1+\sqrt{3})i}{1+i} \\
&&&= \frac{5-\sqrt{3}}{4} + \frac{3-3\sqrt{3}}{4}i, \\
&&& \frac{\sqrt{3}-1}{4} + \frac{1+3\sqrt{3}}{4}i
\end{align*}
\end{questionparts}