Quadratic functional equation

Remarks

Example: on $G = \R^2$, $f(x_1, x_2) = x_1^2 + 3 x_1 x_2$ is a solution, since it is a quadratic form and the cross terms in $f(x + y) + f(x - y)$ cancel. Non-example: $f(x) = x$ on $\R$ is not a solution, since the left side is $2x$ and the right side is $2x + 2y$. If $f$ is a solution, then $B(x, y) = \frac{1}{4}\big(f(x + y) - f(x - y)\big)$ is symmetric and additive in each variable, and $f(x) = B(x, x)$. Conversely, $f(x) = B(x, x)$ is a solution for every symmetric biadditive $B \colon G \times G \to \R$. On $G = \R$ the continuous solutions are $f(x) = cx^2$.