In this lesson, students learn that completing the square can be used to solve any quadratic equation, including equations that involve rational numbers that are not integers. Students notice that the process of completing the square is the same when the equations involve messier numbers as when they have simple integers, but the calculations may be more time consuming and prone to error. An erroranalysis activity highlights some common errors related to completing the square.
Although any equation can be solved by completing the square, equations that are really difficult to solve by this method are not included here. Students will solve such equations when they have access to the quadratic formula. What is important in this lesson is to recognize that putting a quadratic equation in the form of \((x+p)^2 =q\) allows them to solve it, but there are cases in which doing so may not always be the most efficient strategy.
Completing the square for quadratic expressions that are more elaborate encourages students to look for and make use of the same structure that helped them when they were working with less complicated expressions (MP7).
Lesson overview
 13.1 Warmup: Math Talk: Equations with Fractions (5 minutes)

13.2 Activity: Solving Some Harder Equations (20 minutes)
 Includes "Are you Ready for More?" extension problem
 13.3 Activity: Spot Those Errors! (10 minutes)
 Lesson Synthesis
 13.4 Cooldown: How Did We Get Those Solutions? (5 minutes)
Learning goals:
 Express any quadratic equation in the form \((x+p)^2=q\) and solve the equation by finding square roots.
 Generalize (orally) a process for completing the square to express any quadratic equation in the form \((x+p)^2=q\).
 Solve quadratic equations in which the squared term has a coefficient of 1 by completing the square.
Learning goals (student facing):
 Let’s solve some harder quadratic equations.
Learning targets (student facing):
 When given a quadratic equation in which the coefficient of the squared term is 1, I can solve it by completing the square.
Required materials:
 Scientific calculators
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.