1 In this question you must show detailed reasoning.
The quadratic equation \(x ^ { 2 } - 2 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
- Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
- Hence find a quadratic equation with roots \(\alpha + \frac { 1 } { \beta }\) and \(\beta + \frac { 1 } { \alpha }\).
Using the formulae for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\), show that \(\sum _ { r = 1 } ^ { 10 } r ( 3 r - 2 ) = 1045\).