Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \ge 1$,then the value of $\frac{a_{10} - 2a_8}{2a_9}$ is equal to:

For what value of $m$ will one root of $x^2 - 3x + 2m = 0$ be double one root of $x^2 - x + m = 0$?

If one root of the equation $4x^2 - 2x + k - 4 = 0$ is the reciprocal of the other,then the value of $k$ is

The value of $a$ such that the sum of the squares of the roots of the equation $x^2 - (a - 2)x - a + 1 = 0$ assumes the least value is:

If $\alpha$ and $\beta$ are the roots of the equation $7x^{2}-3x-2=0$,then the value of $\frac{\alpha}{1-\alpha^{2}}+\frac{\beta}{1-\beta^{2}}$ is equal to

Let $p$ and $q$ be real numbers such that $p \neq 0, p^3 \neq q$ and $p^3 \neq -q$. If $\alpha$ and $\beta$ are non-zero numbers satisfying $\alpha + \beta = -p$ and $\alpha^3 + \beta^3 = q$,then find the quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.