Problem
ALG-B3-M01-P006 A square after subtraction
#6
★★★☆☆ Level 3 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+4xy\), \(f(0)=0\), \(f(1)=2\). Find \(f\).
Hint. Subtract \(2x^2\).
Put \(g(x)=f(x)-2x^2\). Then \(g(x+y)=g(x)+g(y)\), and \(g(1)=0\). On \(\mathbb Q\), this gives \(g\equiv0\). Hence \(f(q)=2q^2\).
Goal: separate guessing the answer from a complete proof and domain check.