Lesson 13

Completing the Square (Part 2)

13.1: Math Talk: Equations with Fractions (5 minutes)

Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for operations on fractions and solving equations. These understandings help students develop fluency and will be helpful later in this lesson when students will need to use similar computations to solve equations.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Solve each equation mentally.

\(x+x=\frac14\)

\((\frac32) ^2 = x\)

\(\frac35 + x = \frac95\)

\(\frac{1}{12}+x=\frac14\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

13.2: Solving Some Harder Equations (20 minutes)

Activity

In this activity, students encounter equations that contain rational numbers. They learn that, even though the numbers are not as straightforward as in equations they have previously seen, they can still solve the equations by completing the square. The less-friendly numbers and the more elaborate equations make the computations more challenging, so students need to attend to precision (MP6) and mind the steps for rearranging parts of the equations, as well as the signs of the numbers, especially negative numbers.

As students work, they may need to be reminded to rewrite the equation in a different form, isolate a term to make it easier to complete the square, apply the zero product property only when a product is equal to zero, and so on. Make note of common errors or hurdles and address them during whole-class discussion.

Launch

Give students a moment to look at the list of equations and notice how they are like or unlike other equations they have seen and solved before. Invite students to share some observations. They may notice, for instance:

  • \((x-3)(x+1)=5\) does not have 0 on one side, so we cannot use the zero product property right away.
  • \(x^2 + \frac12 x = \frac{3}{16}\) and \(x^2+3x+\frac84=0\) involve fractions.
  • \(x^2+3x+\frac84=0\) has an odd number for the coefficient of the linear term.
  • \((7-x)(3-x)+3=0\) has 0 on one side but the other side is a sum, not a product.
  • \(x^2+1.6x+0.63=0\) has decimals.

Explain to students that each equation can be solved by completing the square and using the same reasoning as when we solved simpler-looking equations earlier.

Consider arranging students in groups of 2. If there is not enough time for students to answer all the questions, consider asking students to look at the list of equations and to choose one they think looks easier (or more familiar) and one that looks harder (or less familiar).

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select at least 3 of the equations to solve. Encourage students to select the problems ahead, and persist with the problems they have chosen for the duration of the task (not reselecting if a problem becomes challenging for them). Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Solve these equations by completing the square.

  1. \((x-3)(x+1)=5\)
  2. \(x^2 + \frac12 x = \frac{3}{16}\)
  3. \(x^2+3x+\frac84=0\)
  4. \((7-x)(3-x)+3=0\)
  5. \(x^2+1.6x+0.63=0\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Show that the equation \(x^2+10x+9=0\) is equivalent to \((x+3)^2+4x=0\).
  2. Write an equation that is equivalent to \(x^2+9x+16=0\) and that includes \((x+4)^2\).
  3. Does this method help you find solutions to the equations? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may have a difficult time getting started given the increased complexity of these problems. Encourage these students to use completing the square to solve a more familiar problem first, such as \(x^2-10x+20=4\), and then to use the structure of the process they followed (MP7) as a guide to help them solve these more challenging problems.

Activity Synthesis

Select students to share their solutions and display their work for all to see. If students found solutions that differ from each other, as time permits, invite these students to describe their thinking to the class, and encourage them to try to identify any errors and determine which solutions are correct. Discuss any common challenges or errors.

Make sure students understand that the process of solving by completing the square is the same whether the numbers in the equations are integers or other rational numbers. Elicit from students a generalization of the process. For example:

  • Start with a quadratic expression in standard form on one side of the equal sign and a number (which could be 0) on the other side.
  • Take half of the coefficient of the linear term and square it. The result is the constant term that would make the quadratic expression a perfect square.
  • Perform the same operations to both sides of the equal sign so the constant term that is on one side is a perfect square that makes the expression a perfect square.
  • Rewrite the expression (in standard form) as a squared factor.
  • Find both values of the factor that, when squared, gives the number on the other side of the equal sign.
Conversing: MLR2 Collect and Display. During the whole-class discussion, listen for and collect language students use to generalize the process of solving by completing the square. Write students’ words and phrases on a visual display, keeping track of each step from start to finish. If necessary, use a specific example to make connections to the generalized steps by highlighting, making annotations, or drawing arrows.  For example, when describing the “coefficient of the linear term” or the “perfect square” in the generalized process, highlight the actual terms in the example. Remind students to borrow language from the display and use it as a resource to draw language from during small-group and whole-class discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

13.3: Spot Those Errors! (10 minutes)

Activity

In this activity, students analyze worked examples of equations solved by completing the square. The work allows them to further develop their understanding of the method when used with rational numbers and to notice common errors. Identifying errors in worked examples is an opportunity to attend to precision (MP6) and to analyze and critique the reasoning of others (MP3).

Launch

Display the four equations in the task statement for all to see:

  1. \(x^2 + 14x= \text-24\)
  2. \(x^2 - 10x + 16= 0\)
  3. \(x^2 + 2.4x = \text-0.8\)
  4. \(x^2 - \frac65 x + \frac15 = 0\)

Arrange students in groups of 3–4. Assign one equation (or more, depending on time constraints) for each group to solve by completing the square. Once group members agree on the solutions, ask them to look for errors in the worked solution for the same equation (as the one they solved).

Engagement: Develop Effort and Persistence. Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2 students to share how they identified the error in the first worked solution. Record their thinking on a display for all to see, and keep the work visible as students continue.
Supports accessibility for: Organization; Attention

Student Facing

Here are four equations, followed by worked solutions of the equations. Each solution has at least one error.

  • Solve one or more of these equations by completing the square.
  • Then, look at the worked solution of the same equation as the one you solved. Find and describe the error or errors in the worked solution.
  1. \(x^2 + 14x= \text-24\)
  2. \(x^2 - 10x + 16= 0\)
  3. \(x^2 + 2.4x = \text-0.8\)
  4. \(x^2 - \frac65 x + \frac15 = 0\)

Worked solutions (with errors):

1.

\(\displaystyle \begin {align} x^2 + 14x &= \text-24\\ x^2 + 14x + 28 &= 4\\ (x+7)^2 &= 4\\ \\x+7 = 2 \quad &\text {or} \quad x+7 = \text-2\\ x = \text-5 \quad &\text {or} \quad x = \text-9 \end{align}\)

2.

\(\displaystyle \begin {align} x^2 - 10x + 16 &= 0\\x^2 - 10x + 25 &= 9\\(x - 5)^2 &= 9\\ \\x-5=9 \quad &\text {or} \quad x-5 = \text-9\\ x=14 \quad &\text {or} \quad x=\text-4 \end{align}\)

3.

\(\displaystyle \begin {align}x^2 + 2.4x &= \text-0.8\\x^2 + 2.4x + 1.44 &= 0.64\\(x + 1.2)^2&=0.64\\x+1.2 &= 0.8\\ x &=\text -0.4 \end{align}\)

4.

\(\displaystyle \begin {align} x^2 - \frac65 x + \frac15 &= 0\\x^2 - \frac65 x + \frac{9}{25} &= \frac{9}{25}\\ \left(x-\frac35\right)^2 &= \frac{9}{25}\\ \\x-\frac35= \frac35 \quad &\text {or} \quad x-\frac35=\text- \frac35\\ x=\frac65 \quad &\text {or} \quad x=0 \end{align}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display the four solutions in the task statement for all to see. Select students to share the errors they spotted and their proposed corrections.

To involve more students in the discussion, after each student presents, consider asking students to classify each error by type (not limited to one type per error) and explain their classification. Here are some examples of types of errors:

  • Careless errors—for example, writing the wrong number or symbol, repeating the same step twice, leaving out a negative sign, or not following directions.
  • Computational errors—for example, adding or subtracting incorrectly or putting a decimal point in the wrong place.
  • Gaps in understanding—for example, misunderstanding of the problem or the rules of algebra, applying an ineffective strategy, or choosing the wrong operation or step.
  • Lack of precision or incomplete communication—for example, missing steps or explanations, incorrect notation, or forgetting labels or parentheses.
Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain the errors they identified and proposed corrections. Invite groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking, and will improve the quality of explanations shared during the whole-class discussion. Make sure to vary who is selected to represent the work of the group, so that students get accustomed to preparing each other to fill that role.
Design Principle(s): Support sense-making; Cultivate conversation

Lesson Synthesis

Lesson Synthesis

To help students generalize the process of solving quadratic equations by completing the square, ask students,

  • “How was the process of completing the square in this lesson different than that in an earlier lesson?” (The numbers in the equations are not as simple, so the calculations may be more involved and time consuming. In some cases, more steps were needed to complete the square, and the process may be more prone to error.)
  • “How was the process of completing the square in this lesson like that in the previous lesson?” (The steps are essentially the same: turn one side into a perfect-square expression, write it as a squared factor, and reason about the value of the factor.)
  • “What are some errors you have seen or have made when the solving process involves multiple steps, especially when the numbers are tricky to work with?” (Some common errors:
    • Only positive solutions are shown. Negative solutions are neglected.
    • Different amounts are added to the two sides of the equal sign.
    • The wrong amount is added so the quadratic expression is not a perfect square.
    • Incorrect calculations or missed steps.)
  • “When might we want to choose to solve quadratic equations by rewriting them in factored form instead of by completing the square? When might it be preferable to complete the square?” (When we could easily see the factors in a quadratic equation, then it makes sense to solve the equation by putting it in factored. When it is not obvious what the factors are, or when the equation involves non-integers, then completing the square may be preferable.)

Emphasize that completing the square is a handy technique that can be used to solve any quadratic equation, but depending on the numbers in the equation, it may or may not be an efficient method.

13.4: Cool-down - How Did We Get Those Solutions? (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Completing the square can be a useful method for solving quadratic equations in cases in which it is not easy to rewrite an expression in factored form. For example, let’s solve this equation:

\(\displaystyle x^2 + 5x - \frac{75}{4}=0\)

First, we’ll add \(\frac{75}{4}\) to each side to make things easier on ourselves.

\(\displaystyle \begin {align} x^2 + 5x - \frac{75}{4}+ \frac{75}{4} &= 0+\frac{75}{4}\\ x^2 + 5x &= \frac{75}{4} \end {align}\)

To complete the square, take \(\frac12\) of the coefficient of the linear term 5, which is \(\frac52\), and square it, which is \(\frac{25}{4}\). Add this to each side:

\(\displaystyle \begin {align}x^2 + 5x + \frac{25}{4} &= \frac{75}{4} + \frac{25}{4}\\x^2 + 5x + \frac{25}{4} &= \frac{100}{4} \end{align}\)

Notice that \(\frac{100}{4}\) is equal to 25 and rewrite it:

\(\displaystyle x^2 + 5x + \frac{25}{4} =25\)

Since the left side is now a perfect square, let’s rewrite it:

\(\displaystyle \left(x+\frac52 \right)^2 = 25\)

For this equation to be true, one of these equations must true:

\(\displaystyle x + \frac52 = 5 \quad \text{or} \quad x + \frac52 = \text-5\)

To finish up, we can subtract \(\frac52\) from each side of the equal sign in each equation.

\(\displaystyle \begin {align} x = 5 - \frac52 \quad &\text{or} \quad x = \text-5 - \frac52\\x = \frac{5}{2} \quad &\text{or} \quad x = \text-\frac{15}{2}\\x = 2\frac12 \quad &\text{or} \quad x = \text-7\frac12 \end{align}\)

It takes some practice to become proficient at completing the square, but it makes it possible to solve many more equations than you could by methods you learned previously.