If the sum of two particular roots of the equation $x^4-4x^3-7x^2+22x+24=0$ is equal to the sum of the remaining two roots,then the sum of the cubes of all the roots of this equation is

The number of solutions of the equation $e^{\sin x} - 2e^{-\sin x} = 2$ is

The roots $\alpha, \beta$ of the equation $x^2-6(k-1)x+4(k-2)=0$ are equal in magnitude but opposite in sign. If $\alpha > \beta$,then the product of the roots of the equation $2x^2-\alpha x+6\beta(\alpha+1)=0$ is

Let $S=\{1, 2, 3, \ldots, 100\}$. Suppose $b$ and $c$ are chosen at random from the set $S$. The probability that $4x^2+bx+c=0$ has equal roots is

The number of positive integers $x$ satisfying the equation $\frac{1}{x} + \frac{1}{x+1} + \frac{1}{x+2} = \frac{13}{12}$ is:

If $x = \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}}$,then find the value of $x$.