Journey with alpha and beta

Algebra Level 4

Let \(\alpha\) and \(\beta\) be the roots of \(x^{2} - 6x -2\) , with \(\alpha > \beta\). If \(a_{n} = \alpha^{n} - \beta^{n}\) for \(n>1\), then find the value of \(\dfrac{a_{10} - 2a_{8}}{2a_{9}}\).