If $\alpha \neq \beta$,$\alpha^2 = 5\alpha - 3$,and $\beta^2 = 5\beta - 3$,then find the equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.

If the difference of the roots of $x^2 - px + 8 = 0$ is $2$,then the value of $p$ is

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - a(x + 1) - b = 0$,then $(\alpha + 1)(\beta + 1) = $

If $\alpha, \beta$ are the roots of $x^2+ax+2=0$ and $\frac{1}{\alpha}, \frac{1}{\beta}$ are the roots of $x^2-bx+c=0$,then $\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right) = $

If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation $x^2 + bx + c = 0,$ where $c < 0 < b,$ then

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