Problem
ALG-B3-M01-P011 Quadratic extra term with normalization
#11
★★★★☆ Level 4 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+2xy\), and \(f(2)=4\). Find \(f\).
Hint. Subtract \(x^2\).
Put \(g(x)=f(x)-x^2\). Then \(g(x+y)=g(x)+g(y)\). Also, \(g(2)=f(2)-4=0\), so \(g(1)=0\). On \(\mathbb Q\), \(g(q)=0\), hence \(f(q)=q^2\). Check: \((x+y)^2=x^2+y^2+2xy\).
Goal: separate guessing the answer from a complete proof and domain check.