If ${x_1}, {x_2}, {x_3}$ are distinct roots of the equation $ax^2 + bx + c = 0$,then:

The number of integers satisfying the inequality $\sqrt{\log_3(x) - 1} + \frac{\frac{1}{2}\log_3(x^3)}{\log_3(\frac{1}{3})} + 2 > 0$ is

If for a positive integer $n$,the quadratic equation $x(x + 1) + (x + 1)(x + 2) + \dots + (x + n - 1)(x + n) = 10n$ has two consecutive integral solutions,then $n$ is equal to:

If $\alpha, \beta$ and $\gamma$ are the roots of $x^3 + px + q = 0$,then the value of $\alpha^3 + \beta^3 + \gamma^3$ is equal to

The roots of $|x - 2|^2 + |x - 2| - 6 = 0$ are

Let $\alpha, \alpha^2$ be the roots of $x^2 + x + 1 = 0$. Then the equation whose roots are $\alpha^{31}, \alpha^{62}$ is: