Question

The quadratic polynomial $P ( x )= ax ^2+ bx + c$ has two different zeroes including -2 . The quadratic polynomial $Q ( x )= ax ^2+ cx + b$ has two different zeroes including 3 . If $\alpha$ and $\beta$ be the other zeroes of $P ( x )$ and $Q ( x )$ respectively, then find the value of $\frac{\alpha}{\beta}$.

Answer 1

$S _{ p }+ P _{ q }=0 \left[ S _{ p } \rightarrow \operatorname{Sum} \text { of } P ( x ) \text { and } P _{ q } \rightarrow \text { Product of } P ( x )\right] $
$\text { and } P _{ p }+ S _{ q }=0 $
$\therefore \alpha-2+3 \beta=0 \Rightarrow \alpha+3 \beta=2 $....(1)
$\text { and } -2 \alpha+3+\beta=0 \Rightarrow-2 \alpha+\beta=-3 \ldots \ldots .(2)$
$\text { From (1) and (2), we get }$
$\alpha=\frac{11}{7} \text { and } \beta=\frac{1}{7} $
$\frac{\alpha}{\beta}=\frac{11}{7} \cdot \frac{7}{1}=11 $