The roots of the equation $x^{2}+px+q=0$ are equal if:

The diagram shows the graph of $y = ax^2 + bx + c$. Then:

Solve the given two equations and select the correct option.
$I.$ $\frac{12}{\sqrt{x}} - \frac{23}{\sqrt{x}} = 5\sqrt{x}$
$II.$ $\frac{\sqrt{y}}{12} - \frac{5\sqrt{y}}{12} = -\frac{1}{\sqrt{y}}$

When $a = -5, b = -6$ and $c = 10$,find the value of $\frac{a^{3} + b^{3} + c^{3} - 3abc}{ab + bc + ca - a^{2} - b^{2} - c^{2}}$.

If $\alpha, \beta$ are the roots of ${x^2} + px + q = 0$ and $\alpha + h, \beta + h$ are the roots of ${x^2} + rx + s = 0$,then

If $\alpha, \beta$ are the roots of $x^2 - px + q = 0$ and $\alpha', \beta'$ are the roots of $x^2 - p'x + q' = 0$,then the value of $(\alpha - \alpha')^2 + (\beta - \alpha')^2 + (\alpha - \beta')^2 + (\beta - \beta')^2$ is