Mathematics - Quadratic Equation Question with Solution | TestHub

Since, $\alpha$ and $\beta$ are the roots of the equation $x^2-2x-1=0$, then Sum of roots, $\alpha +\beta =2$ and product of the roots $\alpha \beta =-1$ Since, $(\alpha +\beta )={\alpha }^2+{\beta }^2+2\alpha \beta$ $\Rightarrow 4={\alpha }^2+{\beta }^2-2$ $\Rightarrow {\alpha }^2+{\beta }^2=6$ Now, ${\alpha }^2{\beta }^{-2}+{\alpha }^{-2}{\beta }^2=\frac{{\alpha }^2}{{\beta }^2}+\frac{{\beta }^2}{{\alpha }^2}=\frac{{\alpha }^4+{\beta }^4}{(\alpha \beta {)}^2}$ $\Rightarrow {\left({\alpha }^2+{\beta }^2\right)}^2=6^2$ $\Rightarrow {\alpha }^4+{\beta }^4+2{\alpha }^2{\beta }^2=36$ $\Rightarrow {\alpha }^4+{\beta }^4+2=36$ $\Rightarrow {\alpha }^4+{\beta }^4=34\dots ..(\mathrm{i})$ $\Rightarrow \frac{{\alpha }^4+{\beta }^4}{(\alpha \beta {)}^2}=\frac{34}{(-1{)}^2}=34$ [Putting value of ${\alpha }^4+{\beta }^4=34$ from Equation (i) $]$