If $\alpha^2 = 5\alpha - 3$ and $\beta^2 = 5\beta - 3$ where $\alpha \neq \beta$,what is the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$?

If the sum of the roots of the equation $ax^2 + bx + c = 0$ is equal to the sum of the reciprocals of their squares,then $bc^2, ca^2, ab^2$ will be in

Two roots of the cubic equation $x^3 + 3x^2 + kx + 12 = 0$ are real and unequal but have the same absolute value. Find the value of $k$.

If the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ is least for an appropriate value of the variable parameter $a$,then the value of $a$ will be

Let $\alpha, \beta$ be the roots of the equation $x^2-ax-b=0$ with $\operatorname{Im}(\alpha) < \operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5 \sqrt{7} i, P_4=-3 \sqrt{7} i, P_5=11 \sqrt{7} i$ and $P_6=45 \sqrt{7} i$,then $|\alpha^4+\beta^4|$ is equal to . . . . . .

For the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,if $\alpha$ and $\beta$ are the roots,then $\alpha^2 + \beta^2 = \dots$