Value of Expression Involving Roots of Quadratic Equation

To find the value of the given expression, we use Newton's Sums for quadratic equations.

Given the quadratic equation:
$x^2 - 6x - 2 = 0$

Since $\alpha$ and $\beta$ are the roots of this equation, they satisfy:
$\alpha^2 - 6\alpha - 2 = 0 \implies \alpha^2 - 2 = 6\alpha$
$\beta^2 - 6\beta - 2 = 0 \implies \beta^2 - 2 = 6\beta$

Multiplying both sides of the equations by $\alpha^{n-2}$ and $\beta^{n-2}$ respectively for $n \ge 2$:
$\alpha^n - 2\alpha^{n-2} = 6\alpha^{n-1}$
$\beta^n - 2\beta^{n-2} = 6\beta^{n-1}$

Subtracting the second equation from the first:
$(\alpha^n - \beta^n) - 2(\alpha^{n-2} - \beta^{n-2}) = 6(\alpha^{n-1} - \beta^{n-1})$

Given that $a_n = \alpha^n - \beta^n$, we can rewrite this recurrence relation as:
$a_n - 2a_{n-2} = 6a_{n-1}$

Substituting $n = 10$ into the relation:
$a_{10} - 2a_8 = 6a_9$

Now, we evaluate the required expression:
$\frac{a_{10} - 2a_8}{2a_9} = \frac{6a_9}{2a_9} = 3$

Thus, the correct option is (C).