Find all functions $f$ for which $f(x-y) = f(x) + f(y) - 2xy$ for all real numbers $x$ and $y$.

Question

Find all functions $f$ for which $f(x-y) = f(x) + f(y) - 2xy$ for all real numbers $x$ and $y$.

I have tried assume $f(x) = n^2$, the equality holds. How can I find other functions that satisfy the equality?

Answer 1

Let $x=0, y=0$. We get $f(0)=f(0)+f(0)$ which means $f(0)=0$.

Now let $y=x$. We get $0=f(0)=f(x)+f(x)-2x^2$. From here, $2f(x)=2x^2$, or $f(x)=x^2$ is the only solution.