The problem of solving systems of polynomial equations is ubiquitous throughout science and engineering. The mathematical subject of numerical algebraic geometry consists of a collection of approaches for numerically solving polynomial systems with one foundational technique being homotopy continuation. This short manuscript summarizes using homotopy continuation on two different problems. In the first problem, homotopy continuation is used to approximate a critical parameter value where two solutions of a parameterized differential equation merge together. In the second problem, homotopy continuation is used to compute critical points of a sum of squares best fit function for given data.