1 The quadratic equation
$$5 x ^ { 2 } - 7 x + 1 = 0$$
has roots \(\alpha\) and \(\beta\).
- Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
- Show that \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha } = \frac { 39 } { 5 }\).
- Find a quadratic equation, with integer coefficients, which has roots
$$\alpha + \frac { 1 } { \alpha } \quad \text { and } \quad \beta + \frac { 1 } { \beta }$$
(5 marks)