Question

if the given polynomial $x^2-2x-\sqrt{5}$. Find value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$

Answer 1

Given: $\alpha$ and $\beta$ are roots of the equation: $x^2-2x-\sqrt{5}$

So, $\alpha +\beta =2$

& $\alpha \beta =-\sqrt{5}$

Now, $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }$

$=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }$

$=\frac{(2{)}^2+2\sqrt{5}}{-\sqrt{5}}$

$=-(\frac{4}{\sqrt{5}}+2)$

So, $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=-(\frac{4}{\sqrt{5}}+2)$