2022 VJC P1 Q5

Do not use a calculator in answering this question.

The complex number \(z\) satisfies the equation

\({{z}^{2}}-\left( 4+\mathbf{i} \right)z+2\left( \mathbf{i}-t \right)=0\)

where \(t\) is a real number. It is given that one root is of the form \(k-k\,\mathbf{i}\), where \(k\) is real and positive. Find \(t\) and \(k\), and the other root of the equation.[7]