If the difference between the roots of the equation $x^2 + ax + b = 0$ is equal to the difference between the roots of the equation $x^2 + bx + a = 0$ $(a \neq b)$,then:

The roots of the quadratic equation $3x^2 - px + q = 0$ are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with a common difference $d = \frac{3}{2}$. If the sum of the first $11$ terms of this arithmetic progression is $88$,then $q - 2p$ is equal to:

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

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-ax^2+bx-c=0$,then $\Sigma \alpha^2(\beta+\gamma) = $

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 the harmonic mean of the roots of the equation $\sqrt{2} x^2 - bx + (8 - 2\sqrt{5}) = 0$ is $4$,then the value of $b$ is