Solve the given two equations and select the correct answer from the given options.
$I.$ $x^{2} = 64$
$II.$ $2y^{2} + 25y + 72 = 0$

The roots of the equation $3 a^{2} x^{2}-a b x-2 b^{2}=0$ are

If the roots of the equation $x^2 + x + 1 = 0$ are $\alpha$ and $\beta$,and the roots of the equation $x^2 + px + q = 0$ are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$,then $p$ is equal to:

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0$ where $0 < \theta < 45^\circ$,and $\alpha < \beta$. Then $\sum_{n=0}^\infty (\alpha^n + \frac{(-1)^n}{\beta^n})$ is equal to

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

Number of integral values of $a$ for which the smaller root of the quadratic equation $x^2 - 2ax - 4 + a^2 = 0$ is smaller than $1$ and the larger root is greater than $6$ is