If $\alpha$ and $\beta$ are roots of the equation $x^2 + px + \frac{3p}{4} = 0,$ such that $|\alpha - \beta| = \sqrt{10},$ then $p$ belongs to the set

If $\alpha$ and $\beta$ are roots of $x^2 - 3x + 1 = 0$,then the equation whose roots are $\frac{1}{\alpha - 2}$ and $\frac{1}{\beta - 2}$ is

Solve the given two equations and select the correct option.
$I.$ $\sqrt{500} x + \sqrt{402} = 0$
$II.$ $\sqrt{360} y + \sqrt{200} = 0$

Solve the given two equations and select the correct option.
$I.$ $63x^2 + 172x + 117 = 0$
$II.$ $30y^2 + 162y + 216 = 0$

Let $a \ne b, c \ne 0$. If the equations $x^2 + ax + bc = 0$ and $x^2 + bx + ac = 0$ have a common root,then:
Statement $-1$: The equation of the other roots is $x^2 + cx + ab = 0$.
Statement $-2$: $a + b + c = 0$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + ax + b = 0$,then the value of $\alpha^3 + \beta^3$ is equal to: