If $a=b=c,$ then find the value of $\frac{(a+b+c)^{2}}{a^{2}+b^{2}+c^{2}}.$

Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2}-6x=7$
$II.$ $2y^{2}+13y+15=0$

One root of the quadratic equation $x^{2}-5x+6=0$ is $3$. Find the other root.

If the roots of the equation $lx^2 + nx + n = 0$ are in the ratio $p:q$,then $\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} + \sqrt{\frac{n}{l}} = $

If $\alpha, \beta$ are the roots of $x^2 + px + 1 = 0$ and $\gamma, \delta$ are the roots of $x^2 + qx + 1 = 0$,then $q^2 - p^2$ =

The harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is