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Completing the square

Several of the explanations on this website require a bit of algebra to understand. “Completing the square” [wikipedia] refers to taking an expression of the form $a x^{2} + b x + c$ and, though algebraic manipulation, ending up with an expression of the form $a (x + d)^{2} + e$ . Completing the square is an useful algebra trick to know since it arises often when multiplying likelihoods by priors, especially in the context of the Normal distribution.

In this reference tutorial, we will demonstrate how to complete the square in both univariate (scalar) and multivariate (matrix) contexts. Incidentally, this article also contains the derivation of the posterior distribution of the mean of the normal distribution when the variance is known in both univariate and multivariate contexts.