Let $\alpha, \beta$ be roots of $x^2+\sqrt{2}x-8=0$. If $U_n = \alpha^n + \beta^n$,then $\frac{U_{10} + \sqrt{2}U_9}{2U_8}$ is equal to ............

If $\alpha, \beta, \gamma$ are the roots of $x^3+4x+1=0$,then the equation whose roots are $\frac{\alpha^2}{\beta+\gamma}, \frac{\beta^2}{\gamma+\alpha}, \frac{\gamma^2}{\alpha+\beta}$ is

For the equation $\frac{1}{x + a} - \frac{1}{x + b} = \frac{1}{x + c}$,if the product of the roots is zero,what is the sum of the roots?

If $\alpha, \beta$,where $\alpha < \beta$,are the roots of the equation $\lambda x^{2} - (\lambda + 3)x + 3 = 0$ such that $\frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}$,then the sum of all possible values of $\lambda$ is:

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

If $\alpha, \beta, \gamma$ are the roots of $x^3+2x^2-3x-1=0$,then $\alpha^{-2}+\beta^{-2}+\gamma^{-2}=$