3. The quadratic equation
$$2 x ^ { 2 } - 5 x + 7 = 0$$
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
- \(\alpha ^ { 2 } + \beta ^ { 2 }\)
- \(\alpha ^ { 3 } + \beta ^ { 3 }\)
- find a quadratic equation that has roots
$$\frac { 1 } { \alpha ^ { 2 } + \beta } \text { and } \frac { 1 } { \beta ^ { 2 } + \alpha }$$
giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers to be determined.