If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+3ax+2a^{2}=0$ and if $\alpha^{2}+\beta^{2}=5$,find the value of $a$.

Solve the given two equations and select the correct answer from the given options.
$I.$ $81x^2 - 9x - 2 = 0$
$II.$ $56y^2 - 13y - 3 = 0$

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

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

If $x+\frac{1}{x}=3,$ then the value of $x^{8}+\frac{1}{x^{8}}$ will be

If $\alpha, \beta$ are the roots of the equation $a x^{2}+b x+b=0,$ then $\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{b}{a}}=$