If $\alpha$ and $\beta$ are the roots of $9x^2 + 6x + 1 = 0$,then the equation with the roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is

Let $\alpha$ and $\beta$ be the roots of $x^2 - 6x - 2 = 0$,where $\alpha > \beta$. If $a_n = \alpha^n - \beta^n$ for all $n \geq 1$,then what is the value of $\frac{a_{10} - 2a_8}{2a_9}$?

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 2x + 3 = 0$,then find the equation whose roots are $\frac{\alpha - 1}{\alpha + 1}$ and $\frac{\beta - 1}{\beta + 1}$.

If $\alpha, \beta$ are the roots of the quadratic equation $x^{2}+a x+b=0, (b \neq 0),$ then the quadratic equation whose roots are $\alpha-\frac{1}{\beta}$ and $\beta-\frac{1}{\alpha}$ is:

If the roots of the equation $5x^2 - 7x + k = 0$ are reciprocal to each other,then the value of $k$ is:

Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3}x-16=0$,and $\gamma$ and $\delta$ be the roots of $x^2+3x-1=0$. If $P_{n}=\alpha^{n}+\beta^{n}$ and $Q_{n}=\gamma^{n}+\delta^{n}$,then $\frac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to