Question

The quadratic $a{x}^2+bx+c=0$ has two different roots including the root -2 . The equation $a{x}^2+cx+b=0$ has two different roots including the root 3 . The absolute value of the product of the four roots of the two equation expressed in lowest rational is $\left(\frac{p}{q}\right)$. Find $(p+q)$.

Answer 1

Answer: 115

Now sum of roots of $1^{\text{st }}+$ product of $2^{\text{nd }}=0$
$(\alpha -2)+3\beta =0$ ....(3)
and product of $1^{\text{st }}+$ ssum of $2^{\text{nd }}=0$
$-(2a)+3+\beta =0$ ....(4)
$-2(2-3\beta )+3+\beta =0$
$7\beta =1\Rightarrow \beta =\frac{1}{7}$
$\alpha =2-\frac{3}{7}=\frac{11}{7}$
Product $=\frac{11}{7}\cdot \frac{1}{7}(-2)(3)=\left|\frac{-66}{49}\right|=\frac{66}{49}$
$p+q=115$