Relation between roots of two different quadratic equations:

Question

For example, $p,q$ are roots of $ax^2+bx+c=0$ and $r,s$ are roots of $cx^2+dx+e=0$. Then, my question is, can we find the relation between (if $p,q>r,s$ or $p,q \le r,s$, etc.) without actually finding the roots of the given quadratic equations?

Answer 1

Setting $p$ and $q$ in the first equation we have $$(p-q)(ap+aq+b)=0$$ so we get $$p=q$$ or $$a(p+q)+b=0$$ the same for the second equation