Without using a calculator, use the formula for the solution of a quadratic equation to solve
$$( 2 - \mathrm { i } ) z ^ { 2 } + 2 z + 2 + \mathrm { i } = 0$$
Give your answers in the form \(a + b \mathrm { i }\).
The complex number \(w\) is defined by \(w = 2 \mathrm { e } ^ { \frac { 1 } { 4 } \pi \mathrm { i } }\). In an Argand diagram, the points \(A , B\) and \(C\) represent the complex numbers \(w , w ^ { 3 }\) and \(w ^ { * }\) respectively (where \(w ^ { * }\) denotes the complex conjugate of \(w\) ). Draw the Argand diagram showing the points \(A , B\) and \(C\), and calculate the area of triangle \(A B C\).