Question 8 - A-Level Maths

The quadratic equation \(x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha\) and \(\beta\).
1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 4\).
3. Hence find a quadratic equation which has roots \(\alpha ^ { 2 }\) and \(\beta ^ { 2 }\).
The cubic equation \(x ^ { 3 } - 12 x ^ { 2 } + a x - 48 = 0\) has roots \(p , 2 p\) and \(3 p\).
1. Find the value of \(p\).
2. Hence find the value of \(a\).