Question

The quadratic $ax ^2+ bx + c =0$ has two different roots including the root -2 . The equation $ax ^2+ cx + b =0$ 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

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 $2^{\text {nd }}=0$
$-(2 a)+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 $=\frac{11}{7} \cdot \frac{1}{7}(-2)(3)=\left|\frac{-66}{49}\right|=\frac{66}{49}$
$p + q =115$