# Completing the Square

#### This operation, while useful in elementary algebra, also arises frequently when manipulating Gaussian random variables. I review and document both the univariate and multivariate cases.

In elementary algebra, we learned about *completing the square*. This operation has many uses. For most people, the first use of completing the square is finding roots of quadratic equations, and the operation is one way of re-deriving the quadratic formula. I have recently found completing the square useful because it also arises when manipulating Gaussian random variables, since every normal distribution is the exponential of a quadratic function. This post is a review of the univariate case and an extension to the multivariate case.

## Univariate

To review, given a variable $x \in \mathbb{R}$, we want to write a quadratic polynomial

in the form

This second form is often easier to work with. For example, the real roots of the polynomial are immediately obvious from $(2)$. To see how to complete the square, let’s expand the squared term in $(2)$,

And now we just solve for $d$ and $k$,

The reason this trick is called “completing the square” is because it can be viewed as adding a term to $(1)$ such that you can square the equation. For example,

Putting our results together, we get

## Multivariate

Now let’s consider the multivariate extension. Let $\mathbf{x}$ and $\mathbf{b}$ be $d$-dimensional vectors, and let $M \in \mathbb{R}^{d \times d}$ be a symmetric invertible matrix. Then

We can easily verify this by multiplying out the quadratic form,

Of course, the $\mathbf{b}^{\top} M^{-1} \mathbf{b}$ term cancels with the one in $(4)$, and we’re done.