How does the cross multiplication of Quadratic Equation works?
If: $$f_1\left(x\right)=a_1x^2+b_1x+c_1=0$$
and: $$f_2\left(x\right)=a_2x^2+b_2x+c_2=0$$
have a common root, let's say, $\alpha$, then by the method of cross multiplication:
$$\frac{\alpha ^2}{b_1c_2-b_2c_1}=\frac{\alpha }{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}$$
How does that work??
The vector $v = (\alpha^2, \alpha, 1)$ is orthogonal to both $w_1=(a_1,b_1,c_1)$ and $w_2=(a_2,b_2,c_2)$ and assuming that $f_1$ and $f_2$ are independent this implies that $v$ is a multiple of the cross product $w_1 \times w_2$. That's precisely what these equalities express.
I solved the question (for those who wonder how this comes out : )
We have:
$$f_1\left(x\right)=a_1x^2+b_1x+c_1=0$$
$$f_2\left(x\right)=a_2x^2+b_2x+c_2=0$$
Multiply $f_1\left(x\right)$ by $a_2$ and $f_2\left(x\right)$ $a_1$ then use elimination method to obtain one part of the answer.
Then mulitply $f_1\left(x\right)$ by $b_2$ and $f_2\left(x\right)$ by $b_1$ and similarly use elimination method to obtain the second part and then equate all of them simultaneously.