If $x = \sqrt[3]{{\sqrt 2 + 1}} - \sqrt[3]{{\sqrt 2 - 1}}$,then find the value of ${x^3} + 3x$.

Let $p, q, r \in \mathbb{R}$ and $r > p > 0$. If the quadratic equation $px^2 + qx + r = 0$ has two complex roots $\alpha$ and $\beta$,then what is the value of $|\alpha| + |\beta|$?

If $3$ is a root of the equation $x^2 + kx - 24 = 0$,then $3$ is also a root of which of the following equations?

If ${x^2} + {y^2} = 25$ and ${xy} = 12$,then the possible values of $x$ are:

Let $p(x) = x^2 - 5x + a$ and $q(x) = x^2 - 3x + b$,where $a$ and $b$ are positive integers. Suppose $\text{HCF}(p(x), q(x)) = x - 1$ and $k(x) = \text{LCM}(p(x), q(x))$. If the coefficient of the highest degree term of $k(x)$ is $1$,then the sum of the roots of $(x - 1) + k(x)$ is:

The number of solutions of the equation $\sqrt{3x^2+x+5} = x-3$ is