Question

If $\alpha$ and $\beta$ use the zeroes of the polynomial $x^2+px+q$, then find the value of $[\frac{\alpha }{\beta }+2]\times [\frac{\beta }{\alpha }+2]$.

Answer 1

Given equation: $x^2+px+q$

So, $a=1,b=p$ and $c=q$

$[\frac{\alpha }{\alpha }+2][\frac{\beta }{\alpha }+2]$

$=1+2\left(\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }\right)+4$

$=2\left(\frac{{(\alpha +\beta )}^2-2\alpha \beta }{\alpha \beta }\right)+5$

$\alpha +\beta =\frac{-b}{a},\alpha \beta =\frac{c}{a}$

$=2\left(\frac{{\left(\frac{-b}{a}\right)}^2-2\frac{c}{a}}{\frac{c}{a}}\right)+5$

$=2\left(\frac{\frac{b^2-2ac}{a^2}}{\frac{c}{a}}\right)+5$

$=2\left(\frac{b^2-2ac}{ac}\right)+5$

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

Since, $a=1,b=p$ and $c=q$

$\Rightarrow [\frac{\alpha }{\alpha }+2][\frac{\beta }{\alpha }+2]=2(\frac{p^2}{q}-2)+5$