Problem
ALG-B3-M02-P008 Quadratic extra term
#8
★★★★☆ Level 4 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+2xy\), and \(f(1)=4\). Find \(f\).
Hint. Subtract \(x^2\).
Let \(g(x)=f(x)-x^2\). Then \(g(x+y)=g(x)+g(y)\), and \(g(1)=3\). On \(\mathbb Q\), \(g(q)=3q\). Hence \(f(q)=q^2+3q\). Verification follows from \((x+y)^2=x^2+y^2+2xy\).
Goal: choose the right first substitution and then verify the found function.