Problem
ALG-B3-M05-P012 A Quadratic Correction on the Rationals
#12
★★★★☆ Level 4 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) such that \(f(x+y)+f(x-y)=2f(x)+2y^2\) for all \(x,y\).
Subtract \(x^2\).
Let \(g(x)=f(x)-x^2\). Then \(g(x+y)+g(x-y)=2g(x)\). On \(\mathbb Q\), this midpoint equation gives \(g(x)=ax+b\). Hence \(f(x)=x^2+ax+b\), where \(a,b\in\mathbb Q\). Direct checking works.
Practises subtracting a known quadratic part.