(i)[3]
Show all working and, without using a calculator, solve the equation $z^2 + (2\sqrt{6})z + 8 = 0$, writing your answers in the form $x + iy$, where $x$ and $y$ are exact real values.
(ii)[1]
Draw an Argand diagram and show the points that represent the roots.
(iii)[3]
The roots are represented by the points $A$ and $B$, with $O$ as the origin. Determine angle $AOB$.
(iv)[1]
Prove, from the given information, that triangle $AOB$ is equilateral.