This article explains how explorations into the quadratic formula can offer students opportunities to learn about the structure of algebraic expressions. In this article, the author leverages the graphical interpretation of the quadratic formula and describes an activity in which students derive the quadratic formula by quantifying the symmetry of a parabola. By the end of the activity, students are equipped to use the quadratic formula to solve quadratic equations and are prepared to reason about the structure of the quadratic formula. This activity can be used with algebra 1 students to introduce the quadratic formula or more advanced algebra students to enhance their understanding. A student activity sheet accompanies this article.