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  "body": "\n## Definition\n\nLet $G$ be an [[abelian group|082GHNPJ53]], written additively.\nThe *quadratic functional equation* on $G$ is the [[functional equation|SRAN4EKWS0]]\n$$f(x + y) + f(x - y) = 2 f(x) + 2 f(y) \\quad \\text{for all } x, y \\in G,$$\nfor an unknown [[function|0EPAT51DXQ]] $f \\colon G \\to \\R$.\nA solution is called a *quadratic function* on $G$.\n\n## Remarks\n\nExample: 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|0RBQS1JJCR]] and the cross terms in $f(x + y) + f(x - y)$ cancel.\nNon-example: $f(x) = x$ on $\\R$ is not a solution, since the left side is $2x$ and the right side is $2x + 2y$.\nIf $f$ 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 $f(x) = B(x, x)$.\nConversely, $f(x) = B(x, x)$ is a solution for every symmetric biadditive $B \\colon G \\times G \\to \\R$.\nOn $G = \\R$ the continuous solutions are $f(x) = cx^2$.\n",
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