Lesson 19

By now, students are familiar with the quadratic formula and how it can be used to solve quadratic equations. Students started to recognize when the quadratic formula is useful and when another method might be preferred. They also saw some errors that are commonly made when applying the formula. Up to this point, however, students simply took the formula for granted and have not considered how it produces solutions to any quadratic equations. That work takes place in this lesson.

Students learn that the quadratic formula came 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 \(ax^2+bx+c=0\).

To prepare students to complete the square with \(a\), \(b\), and \(c\) remaining as letters, students first transform perfect squares from factored form into standard form, but without evaluating anything. For example, they rewrite \((3x-4)^2\) as \((3x)^2 + 2(3x)(\text-4) + (\text-4)^2\), and then \((kx +m)^2\) as \((kx)^2 + 2(kx)m + m^2\). 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 \(ax^2+bx+c\).

There are different ways to derive the quadratic formula. The path chosen here involves temporarily replacing the \(kx\) in \((kx)^2 + 2(kx)m\) with a single letter, say \(Q\), so the expression for which we are completing the square is a monic quadratic expression: \(Q^2+2Qm\). 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).