Lesson 19

Deriving the Quadratic Formula

Lesson Narrative

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).

Learning Goals

Teacher Facing

  • Explain (in writing) the steps used to derive the quadratic formula.
  • Explain (orally and in writing) how the solutions obtained by completing the square are expressed by the quadratic formula.
  • Understand that the quadratic formula can be derived by generalizing the process of completing the square.

Student Facing

  • Let’s find out where the quadratic formula comes from.

Learning Targets

Student Facing

  • I can explain the steps and complete some missing steps for deriving the quadratic formula.
  • I know how the quadratic formula is related to the process of completing the square for a quadratic equation $ax^2+bx+c=0$.

CCSS Standards


Building Towards

Glossary Entries

  • quadratic formula

    The formula \(x = {\text-b \pm \sqrt{b^2-4ac} \over 2a}\) that gives the solutions of the quadratic equation \(ax^2 + bx + c = 0\), where \(a\) is not 0.