If $x = 2 + 2^{2/3} + 2^{1/3}$,then $x^3 - 6x^2 + 6x = $

If $a+b+c=0,$ then find the value of $\frac{1}{b^{2}+c^{2}-a^{2}}+\frac{1}{c^{2}+a^{2}-b^{2}}+\frac{1}{a^{2}+b^{2}-c^{2}}.$

If $\alpha, \beta$ are the roots of $x^2 - px + q = 0$ and $\alpha', \beta'$ are the roots of $x^2 - p'x + q' = 0$,then the value of $(\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2$ is

For what value of $k$ will the equation $x^2 - (3k - 1)x + 2k^2 + 2k - 11 = 0$ have equal roots?

Find the number of natural number solutions for the equation $x_1 + x_2 = 100$,such that $x_1$ and $x_2$ are not multiples of $5$.

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$ where $ac \neq 0$,then $P(x) \cdot Q(x) = 0$ has at least