Completing the Square Methods & Practice

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SWBAT solve quadratics by completing the square.

Big Idea

Students have the opportunity to complete the square using a variety of methods. They apply what they have learned about completing the square to solve a real world problem.


10 minutes

The purpose of today's class is to give students some time to practice solving quadratics by completing the square.  In the previous lesson they applied this skill to a real world problem and now they will work through some practice problems.  Students can continue to use their own methods for completing the square and then set their vertex form equation equal to zero, or they may want to start their process by moving the constant to the other side of the equation. I like to show different methods to students and then let them decide which way they like best.

I begin class by walking students through an example of completing the square and using the resulting vertex form to identify the vertex and then solve for x-intercepts.  Students may have some confusion about all the different steps of the process.  It might be best to use a quadratic where the a term is -1 so students get the practice of factoring out the negative.  I might use a quadratic like: - x2 + 8x + 9.  I try to be sure students are clear first about completing the square and then about what you can do with vertex form.  This work combines a lot of their learning and they may be confused. This is a good opportunity to pause, take questions, and reinforce key ideas.

Video: Another Method for Completing the Square

10 minutes

In this unit, most students start completing the square using an intuitive method. As the quadratics are about to get more complicated with different values of a, and we become more focused on solving a quadratic rather than putting it in vertex form,  I want to expose them to another method for completing the square.  I have found this Khan Academy video to be helpful.

Original Source:(accessed June 19, 2014) 

I let students watch the video, pause for clarifying points, and let students practice this method with a similar problem after the video. I find many students to be relieved at having some set steps to follow, although I like to emphasize that I have them do it a harder way first so they are understanding the concept behind the steps.


Investigation & Closing

30 minutes

Next, I give students some time to practice!  I usually use a worksheet similar to the Kuta Software worksheet Solving Equations by Completing the Square.  

Ideally, students will work with a partner or in small groups here, sharing their thinking as they work. I really like when students start conversations (unprompted by me) about which method they prefer and how this work is making sense to them. I also encourage them to seek help from each other first, before coming to me.  If there is time, I may have students share out their preferred method and their thoughts about what its advantages are.

As students work, I circulate and look for struggling students. Things that I am watching for:

  • A lot of my students have academic content gaps and may need a refresher or direct instruction about how to work with square roots. I like to discuss the decision to leave the answer in radical form versus finding a decimal approximation. I find many students to do not know that leaving the answer in radical form is often the most accurate answer.
  • Students may struggle with the two roots and the +/- part of taking the square root.
  • Which method students prefer to use and why.



I have students complete an Exit Ticket related to Reflection.  Ask them to complete the follow prompt on an index card:

             Which method do you prefer to Complete the Square? Why?