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 $\dfrac{a_{10} - 2a_{8}}{2a_{9}}$ .