Students learn that the quadratic formula can be derived from the steps of completing the square. Because completing the square always works for solving any quadratic equation, the steps can be generalized into a single formula for solving any equation of the form .

To prepare students to complete the square with , , and remaining as letters, students first transform perfect squares from factored form to standard form without evaluating anything. For example, they rewrite as  and  as . Doing so reinforces and makes explicit the structural connections between the two forms, equipping students to reason in reverse as they complete the square for .

There are different ways to derive the quadratic formula. The path chosen here involves temporarily replacing the in with a new variable, , so the expression for which we are completing the square is easier to work with: . An optional activity in the last lesson on completing the square includes this strategy. If desired, consider using it to familiarize students with the idea of using a temporary placeholder to reason with complicated expressions.

In this lesson, students analyze and complete partially worked-out derivations of the quadratic formula, explaining each step along the way. As they do so, students practice constructing logical arguments (MP3).