6 The equation
$$x ^ { 2 } - 4 x + 13 = 0$$
has roots \(\alpha\) and \(\beta\).
- Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
- Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
- Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
- Find in the form \(p + \mathrm { i } q\) the values of:
- \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
- \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
- Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).