If $x = \sqrt[3]{{\sqrt{2} + 1}} - \sqrt[3]{{\sqrt{2} - 1}}$,then ${x^3} + 3x = $

If $x, y, z$ are real and distinct,then $u = x^2 + 4y^2 + 9z^2 - 6yz - 3zx - 2xy$ is always

If the lengths of two sides of a triangle are the roots of the equation $x^2-2 \sqrt{3} x+2=0$ and the angle between these sides is $\frac{\pi}{3}$,then the perimeter of the triangle is

Suppose,$\alpha$ is the minimum value of $x^2+bx+5$ and $\beta$ is the maximum value of $-x^2+ax+5$. If $[\alpha, \beta]$ is the interval of maximum length for $x$ in which $x^2-10x+24 \leq 0$,then $a^2b^2$ is equal to

Both equations $x^2 + b^2 = 1 - 2bx$ and $x^2 + a^2 = 1 - 2ax$ have exactly one root each,and they share the same root. Then:

The polynomial equation of degree $5$ whose roots are the translates of the roots of $x^5-2x^4+3x^3-4x^2+5x-6=0$ by $-2$ is: