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

For the equation $ax^2 + bx + c = 0$,the roots are $\alpha, \beta$,and for the equation $Ax^2 + Bx + C = 0$,the roots are $\alpha - k, \beta - k$. Then $\frac{B^2 - 4AC}{b^2 - 4ac} = \dots$

If $1, 2, 3$ and $4$ are the roots of the equation $x^4+ax^3+bx^2+cx+d=0$,then $a+2b+c$ is equal to

If $A.M.$ of the roots of a quadratic equation is $8/5$ and $A.M.$ of their reciprocals is $8/7$,then the equation is

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3x^{2} + \lambda x - 1 = 0$ is $15$,then $6(\alpha^{3} + \beta^{3})^{2}$ is equal to

Find the sum of all possible values of $k$ for which the roots of the equation $x^2 + (k + 1)x + \lambda = 0$ are the square of each other.