If the roots of the equation $x^2 + 2mx + m^2 - 2m + 6 = 0$ are equal,then the value of $m$ is:

If $x = \sqrt{7 + 4\sqrt{3}},$ then $x + \frac{1}{x} = $

The coefficient of $x$ in the equation $x^2 + px + q = 0$ was taken as $17$ in place of $13$. Its roots were found to be $-2$ and $-15$. The roots of the original equation are:

Solve the given two equations and select the correct answer from the given options.
$I.$ $18x^2 - 13\sqrt{7}x + 14 = 0$
$II.$ $32y^2 - 19\sqrt{6}y + 9 = 0$

If one real root of the quadratic equation $81x^2 + kx + 256 = 0$ is the cube of the other root,then a value of $k$ is

If $x=332, y=333, z=335,$ then the value of $x^{3}+y^{3}+z^{3}-3xyz$ is