2
2. The quadratic equation
$$5 x ^ { 2 } - 2 x + 3 = 0$$
has roots \(\alpha\) and \(\beta\).
Without solving the equation,
- write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
- determine, giving each answer as a simplified fraction, the value of
- \(\alpha ^ { 2 } + \beta ^ { 2 }\)
- \(\alpha ^ { 3 } + \beta ^ { 3 }\)
- determine a quadratic equation that has roots
$$\left( \alpha + \beta ^ { 2 } \right) \text { and } \left( \beta + \alpha ^ { 2 } \right)$$
giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.