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

If $\frac{\alpha}{\alpha+1}$ and $\frac{\beta}{\beta+1}$ are the roots of the quadratic equation $x^2+7x+3=0$,then the equation having roots $\alpha$ and $\beta$ is

If the sum of the roots of the equation $x^2 + px + q = 0$ is equal to the sum of their squares,then

Let $\alpha, \beta$ be the roots of the equation $x^{2}-6x-2=0$ with $\alpha>\beta$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$,then the value of $\frac{a_{10}-2a_{8}}{2a_{9}}$ is

The values of $b$ and $c$ for which the identity $f(x+1)-f(x)=8x+3$ is satisfied,where $f(x)=bx^2+cx+d$,are

If the equation $x^{2}-cx+d=0$ has roots equal to the fourth powers of the roots of $x^{2}+ax+b=0,$ where $a^{2}>4b,$ then the roots of $x^{2}-4bx+2b^{2}-c=0$ will be