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 $A$ is the $A.M.$ of the roots of the equation $x^2 - 2ax + b^2 = 0$ and $G$ is the $G.M.$ of the roots of the equation $x^2 - 2bx + a^2 = 0,$ then

If $\alpha$ and $\beta$ are the roots of the equation $4x^2 - \sqrt{13}x - 7 = 0$,then what is the value of $|\alpha - \beta|$?

If $\alpha$ and $\beta$ are roots of the equation $x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$ for some $k$,and $\alpha^2 + \beta^2 = 66$,then $\alpha^3 + \beta^3$ is equal to: (in $\sqrt{2}$)

The two roots of the equation $x^3 - 9x^2 + 14x + 24 = 0$ are in the ratio $3 : 2$. Find the roots.

If $\alpha, \beta, \gamma$ are the real roots of the equation $x^3 - 3px^2 + 3qx - 1 = 0$,then find the centroid of the triangle whose vertices are $(\alpha, \frac{1}{\alpha}), (\beta, \frac{1}{\beta})$ and $(\gamma, \frac{1}{\gamma})$.