- The quadratic equation
$$3 x ^ { 2 } + 2 x + 5 = 0$$
has roots \(\alpha\) and \(\beta\).
Without solving the equation,
- find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\)
- show that \(\alpha ^ { 3 } + \beta ^ { 3 } = \frac { 82 } { 27 }\)
- find a quadratic equation which has roots
$$\left( \alpha + \frac { \alpha } { \beta ^ { 2 } } \right) \text { and } \left( \beta + \frac { \beta } { \alpha ^ { 2 } } \right)$$
giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers.