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

If $\alpha$ and $\beta$,$\alpha$ and $\gamma$,$\alpha$ and $\delta$ are the roots of the equations $ax^2 + 2bx + c = 0$,$2bx^2 + cx + a = 0$ and $cx^2 + ax + 2b = 0$ respectively,where $a, b$ and $c$ are positive real numbers,then $\alpha + \alpha^2 = $

If $a, b,$ and $c$ are the roots of $x^3+4x+1=0$,then $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=$

The cubic equation whose roots are thrice to each of the roots of $x^3+2x^2-4x+1=0$ is

Consider the two equations $x^2 - 2ax + b^2 = 0$ and $x^2 - 2bx + a^2 = 0$. The arithmetic mean of the roots of the first equation is equal to what?

The coefficient of $x$ in the equation $x^2 + px + q = 0$ was taken as $17$ in place of $13$. Its roots were found to be $-2$ and $-15$. The roots of the original equation are: