It is given that \(\alpha\) and \(\beta\) are the roots of the equation
$$x ^ { 2 } - 2 x + 4 = 0$$
Without solving this equation, show that \(\alpha ^ { 3 }\) and \(\beta ^ { 3 }\) are the roots of the equation
$$x ^ { 2 } + 16 x + 64 = 0$$
(6 marks)
State, giving a reason, whether the roots of the equation
$$x ^ { 2 } + 16 x + 64 = 0$$
are real and equal, real and distinct, or non-real.
Solve the equation
$$x ^ { 2 } - 2 x + 4 = 0$$
Use your answers to parts (a) and (b) to show that
$$( 1 + \mathrm { i } \sqrt { 3 } ) ^ { 3 } = ( 1 - \mathrm { i } \sqrt { 3 } ) ^ { 3 }$$