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  "body": "\n## Summary\n\nThe quadratic functional equation is the [[functional equation|SRAN4EKWS0]] $f(x + y) + f(x - y) = 2 f(x) + 2 f(y)$.\nThe unknown $f$ maps an abelian group, usually a real vector space, to the real numbers.\nEvery [[quadratic form|0RBQS1JJCR]] solves it, and on $\\R$ the continuous solutions are the functions $f(x) = cx^2$.\nIt is the functional form of the [[parallelogram law|026HPGYGS4]], and it lies behind the [[Jordan–von Neumann theorem|46WGVM5VS6]].\n\n## Explanations\n\n### L1 human\n\nFor $f(x) = x^2$ the equation is the identity $(x + y)^2 + (x - y)^2 = 2x^2 + 2y^2$.\nFor the squared length of vectors it is the parallelogram law: the squares of the two diagonals add up to the squares of the four sides.\nThe equation asks which functions behave like a square in this sense.\nIts solutions are exactly the functions of the form $f(x) = B(x, x)$ for a symmetric biadditive map $B$.\n",
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