If the sum of any two roots of the equation $x^3+p x^2+q x+r=0$ is zero,then

If $\alpha$ and $\beta$ are the real roots of the equation $x^2+ax+b=0$,where $\alpha+\beta=\frac{1}{2}$ and $\alpha^3+\beta^3=\frac{37}{8}$,then find the value of $a-\frac{1}{b}$.

If $x^2 + px + 1$ is a factor of $ax^3 + bx + c$,then:

If $2$ and $6$ are the roots of the equation $ax^2 + bx + 1 = 0$,then the quadratic equation,whose roots are $\frac{1}{2a + b}$ and $\frac{1}{6a + b}$,is:

Let $\alpha$ and $\beta$ be the roots of the equation $5x^{2} + 6x - 2 = 0$. If $S_{n} = \alpha^{n} + \beta^{n}$ for $n = 1, 2, 3, \dots$,then:

If $\alpha, \beta$ are the roots of the equation $x^2-2 \sqrt{3} x+4=0$,then $\alpha^6+\beta^6=$