Question

The quadratic polynomial $P(x)=a{x}^2+bx+c$ has two different zeroes including -2 . The quadratic polynomial $Q(x)=a{x}^2+cx+b$ 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

Answer: 11

$S_p+P_q=0\left[S_p\to \mathrm{Sum}\text{ of }P(x)\text{ and }{P}_q\to \text{ Product of }P(x)\right]$
$\text{ and }{P}_p+S_q=0$
$\therefore \alpha -2+3\beta =0\Rightarrow \alpha +3\beta =2$....(1)
$\text{ and }-2\alpha +3+\beta =0\Rightarrow -2\alpha +\beta =-3\dots \dots .(2)$
$\text{ From (1) and (2), we get }$
$\alpha =\frac{11}{7}\text{ and }\beta =\frac{1}{7}$
$\frac{\alpha }{\beta }=\frac{11}{7}\cdot \frac{7}{1}=11$