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

If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of their squares,then:

Sum of the solutions of the equation $[x^2] - 2x + 1 = 0$ is (where $[.]$ denotes the greatest integer function).

If $\log _{10}(x^{2}-6x+45)=2$,then the values of $x$ are

If the roots of the equation $ax^2 + bx + c = 0$ are $(\alpha - \beta)$ and $(\gamma - \delta)$,and the roots of the equation $Ax^2 + Bx + C = 0$ are $(\alpha + \delta)$ and $(\beta + \gamma)$,then $\left| \frac{a}{A} \right|$ is equal to (where $D_1$ and $D_2$ are the discriminants of the given equations respectively).

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