If $\sin 2 \theta$ and $\cos 2 \theta$ are solutions of $x^2+bx-c=0$,then

Which equation has roots that are the squares of the roots of the equation $ax^2 + bx + c = 0$?

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+a x^2-b x+c=0$,then $\sum \beta^2(\gamma+\alpha) = $

If $\frac{1}{\sqrt{\alpha}}$ and $\frac{1}{\sqrt{\beta}}$ are the roots of the equation $ax^2 + bx + 1 = 0$ $(a \ne 0, a, b \in R)$,then the equation $x(x + b^3) + (a^3 - 3abx) = 0$ has roots

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+a x^2+b x+c=0$,then $(\alpha+\beta-2 \gamma)(\beta+\gamma-2 \alpha)(\gamma+\alpha-2 \beta)=$

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