Complex Numbers - Modulus, Argument, and Operations on Complex Numbers

Do not use a calculator in answering this question.

1. Find the roots of the equation \(x^2(1+\mathrm{i})-4x+(4-4\mathrm{i})=0\), giving your answers in cartesian form \(a+\mathrm{i}b\).[3]
2. The complex numbers \(z\) and \(w\) are such that \(z=1+\mathrm{i}\) and \(w=-1-\mathrm{i}\sqrt{3}\). Find the modulus and argument of \(\frac{z}{w^*}\) exactly.[3]