Suppose $a,b,c,A,B,C$ are real numbers with $a\ne0$ and $A\ne0$ such that for all $x$, $$\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.$$Prove that $$\left|b^2-4ac\right|\le\left|B^2-4AC\right|.$$
Source: Serbia MO 2006 2nd Grade P1
Tags: quadratics, Inequality, algebra, inequalities
Suppose $a,b,c,A,B,C$ are real numbers with $a\ne0$ and $A\ne0$ such that for all $x$, $$\left|ax^2+bx+c\right|\le\left|Ax^2+Bx+C\right|.$$Prove that $$\left|b^2-4ac\right|\le\left|B^2-4AC\right|.$$