Given $\,\left|px^2 +qx +r\right|\leqslant\left|Px^2 +Qx +R\right|\,$ for all real $x$ where $P,Q$ and $R$ are different from $p,q$ and $r$, I wish to find the relation between the roots of these quadratic expressions assuming both of them have real roots.So, their discriminants are both positive.
I'm unable to find where to start with.
How do I mathematically prove this? I can visualise the scenario using a graph but, I don't get where to begin, mathematically.
Can someone help me out to proceed? Thanks in advance.