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Solution

Given equation: $x^2 - 6x - 2 = 0$ with roots $\alpha, \beta$ From this: $\alpha + \beta = 6,\quad \alpha\beta = -2$ Using Newton’s theorem / recurrence for powers: Since roots satisfy equation, $\alpha^n = 6\alpha^{n-1} + 2\alpha^{n-2}$ $\beta^n = 6\beta^{n-1} + 2\beta^{n-2}$ Subtract: $a_n = \alpha^n - \beta^n = 6a_{n-1} + 2a_{n-2}$ Now use smart manipulation (no long expansion): We need: $\frac{a_{10} - 2a_8}{3a_9}$ Use recurrence: $a_{10} = 6a_9 + 2a_8$ Substitute: $a_{10} - 2a_8 = (6a_9 + 2a_8) - 2a_8 = 6a_9$ So expression becomes: $\frac{6a_9}{3a_9} = 2$