We show that completing the square for certain polynomials is the same performing as Gaussian Elimination on the associated symmetric matrix $\mathbf{A}$ used in the form $\vec{x^{\rm T}} \mathbf{A} \vec{x}$. We arrive at the Cholesky factorization, a special form of $\mathbf{L}\mathbf{U}$ factorization available when a matrix is symmetric.