If $\alpha, \beta$ are the roots of $(x - a)(x - b) = c$,where $c \neq 0$,then the roots of $(x - \alpha)(x - \beta) + c = 0$ are

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

If $\alpha, \beta$ are roots of the equation $x^{2}+5 \sqrt{2} x+10=0$,$\alpha > \beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $n$,then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{17} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to $....$

$\alpha, \beta, \gamma$ are the roots of the equation $8x^3 - 42x^2 + 63x - 27 = 0$. If $\beta < \gamma < \alpha$ and $\beta, \gamma, \alpha$ are in geometric progression,then the extreme value of the expression $\gamma x^2 + 4\beta x + \alpha$ is

If $\sin \theta$ and $\cos \theta$ are the roots of the equation $ax^2 - bx + c = 0$, then $a, b$ and $c$ satisfy the relation:

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