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 $\alpha$ and $\beta$ are the roots of the quadratic equation $11 x^2 + 12 x - 13 = 0$, then find the value of $\frac{1}{\alpha^2} + \frac{1}{\beta^2}$.

If $\alpha, \beta, \gamma$ are the roots of the equation $4x^3 + 12x^2 - 7x + 165 = 0$ and $\alpha + 5, \beta + 5, \gamma + 5$ are the roots of the equation $ax^3 + bx^2 + cx + d = 0$,then the product of the roots of the second equation is:

The condition that the roots of $x^3-b x^2+c x-d=0$ are in geometric progression is

If the roots of the equation $ax^2 + bx + c = 0$ are real and of the form $\frac{\alpha}{\alpha - 1}$ and $\frac{\alpha + 1}{\alpha}$,then the value of $(a + b + c)^2$ is

For the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,if $\alpha$ and $\beta$ are the roots,then $\alpha^2 + \beta^2 = \dots$