Problem
ALG-B2-M12-P008 Set 2. Reversed quadratic
#8
★★★★★ Level 5 of 5
Let \(Ax^2+Bx+C>0\) for all real \(x\). Prove that \(Cx^2+Bx+A>0\) for all real \(x\).
Hint. Compare the discriminants of the two quadratics.
We have \(A>0\), \(C=P(0)>0\), and \(B^2-4AC<0\). The quadratic \(Cx^2+Bx+A\) has leading coefficient \(C>0\) and the same discriminant, so it is positive for all \(x\).
Mock set 2: the problem is intended for independent method selection without an explicit cue in the statement.