The quadratic equation
$$2 x ^ { 2 } - x + 3 = 0$$
has roots \(\alpha\) and \(\beta\).
Without solving the equation,
- write down the value of \(( \alpha + \beta )\) and the value of \(\alpha \beta\)
- find the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
- find a quadratic equation which has roots
$$\left( 2 \alpha - \frac { 1 } { \beta } \right) \text { and } \left( 2 \beta - \frac { 1 } { \alpha } \right)$$
giving your answer in the form \(p x ^ { 2 } + q x + r = 0\) where \(p , q\) and \(r\) are integers.