Lesson 12

Completing the Square (Part 1)

12.1: Perfect or Imperfect? (5 minutes)

Warm-up

This activity reinforces the meaning of perfect squares and the fact that a perfect square can appear in different forms. To recognize a perfect square, students need to look for and make use of structure (MP7).

Student Facing

Select all expressions that are perfect squares. Explain how you know.

  1. \((x+5)(5+x)\)
  2. \((x+5)(x-5)\)
  3. \((x-3)^2\)
  4. \(x-3^2\)
  5. \(x^2+8x+16\)
  6. \(x^2+10x+20\)

Student Response

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

Activity Synthesis

Display the expressions for all to see. Invite students to share their responses and record them for all to see. For each expression that they consider a perfect square, ask them to explain how they know. For expressions that students believe aren’t perfect squares, ask them to explain why not.

For the last expression, \(x^2+10x+20\), students may reason that it is not a perfect square because:

  • The constant term 20 is not a perfect square. (Most students are likely reason this way.)
  • If 20 is seen as \((\sqrt {20})^2\), then the coefficient of the linear term would have to be \(2\sqrt {20}\), not 10.

Though students have been dealing mostly with rational numbers, the second line of reasoning is also valid and acceptable.

12.2: Building Perfect Squares (10 minutes)

Activity

In this activity, students begin to complete the square. They start by transforming given perfect squares from standard form to factored form, and vice versa. Then, they are given an incomplete expression in standard form that contains only the squared term and linear term. Students need to decide what constant term makes the expression a perfect square, and then write the equivalent expression in factored form. To accomplish these tasks, students must rely on the structure they noticed in an earlier lesson about the relationship between the standard and factored forms of perfect squares (MP7).

Launch

Arrange students in groups of 2. Give students a few minutes of quiet think time and then ask them to discuss their responses with their partner. Follow with a whole-class discussion.

Conversing: MLR2 Collect and Display. Listen for and collect vocabulary, gestures, and diagrams students use to explain how to rewrite perfect squares from standard form to factored form, and vice versa. Write the students’ words on a visual display and update it throughout the remainder of the lesson. Refer back to the visual display during the activity synthesis to reiterate key ideas. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-class discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making 

Student Facing

Complete the table so that each row has equivalent expressions that are perfect squares. 

standard form factored form
1. \(x^2+6x+9\)  
2. \(x^2-10x+25\)  
3.                                   \((x-7)^2\)
4. \(x^2-20x+\underline{\hspace{0.5in}}\) \((x- \underline{\hspace{0.5in}})^2\)
5. \(x^2+16x+\underline{\hspace{0.5in}}\) \((x+ \underline{\hspace{0.5in}})^2\)
6. \(x^2+7x+\underline{\hspace{0.5in}}\) \((x+ \underline{\hspace{0.5in}})^2\)
7. \(x^2+bx+\underline{\hspace{0.5in}}\) \((x+ \underline{\hspace{0.5in}})^2\)

Student Response

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

Anticipated Misconceptions

If students have trouble determining the constant term in standard form, suggest that they draw a rectangular diagram and work backward to determine the factors along the two sides of the rectangle. Afterward, they can find the corresponding value of the constant term.

For example, for the expression \(x^2-20x+ \underline{\hspace{0.5in}}\) we would write: 

  \(x\)  
\(x\) \(x^2\) \(\text-10x\)
  \(\text-10x\)  

Which leads to the the completed diagram:

  \(x\) \(\text-10\)
\(x\) \(x^2\) \(\text-10x\)
\(\text-10\) \(\text-10x\) \(100\)

This diagram represents the expression \(x^2-20x+100\) (in standard form) and the expression \((x - 10)^2\) (in factored form).

Activity Synthesis

Display the incomplete table for all to see. Ask students to complete the missing values or expressions.

Discuss how students knew what numbers or expressions to write in the last four pairs of expressions. Make sure students understand that:

  • If the constant term in the factored form is \(n\), the constant term in standard form is \(n^2\), a squared number.
  • The coefficient of the linear term in standard form is twice of \(n\), or \(2n\).
  • This means that \(n\) is half of the coefficient of the linear term. If the linear term is \(20x\), the \(n\) is 10, and the constant term in standard form is \(10^2\) or 100.
  • If the linear term in standard form is \(px\) (as in the last row of the table), then the constant term in standard form is \(\left(\frac{p}{2}\right)^2\), and the constant term in factored form is \(\frac{p}{2}\).

Explain to students that finding the constant term to add in order to create a perfect square is called “completing the square.” In the next activity, we will look at completing the square as a strategy to solve quadratic equations.

Representation: Internalize Comprehension. Use color and annotations to illustrate student thinking. As students share their conclusions about the relationship between the terms in different forms of perfect squares, scribe their thinking on a visible display. Choose one example from the table. Highlight the coefficient of the linear term in the standard form expression, labeling it \(n\), then highlight the constant term in the standard form expression and write \(\left(\frac{n}{2}\right)^2\). Finally, write \(\frac{n}{2}\) next to the constant term in the factored form expression and highlight it in the same color. Encourage students to select and annotate another example from the table on their own to check for understanding.
Supports accessibility for: Visual-spatial processing; Conceptual processing

12.3: Dipping Our Toes in Completing the Square (20 minutes)

Activity

Earlier, students recognized that an expression in standard form can be written as a perfect square and learned to write it that way. Here, they learn to use that skill to solve quadratic equations.

Launch

Consider keeping students in groups of 2.

Display for all to see the two solution methods in the activity statement. Ask students if the quadratic expression in the original equation is a perfect square. Then, ask them to study the methods and make sense of the steps. Afterward, discuss with students:

  • “How are the two solution methods alike?” (They both involve making the expression on the left a perfect square. They are the steps for solving the same equation, so the solutions are the same.)
  • “How they are different?” (In the first solution, Diego first subtracted 9 from each side, and then added 25 to complete the square. In the second, Mai added 16 to 9 because that is what is needed to make 25.)

Tell students that either method works, but some people prefer the first approach because moving the original constant term to the other side of the equal sign (the right side, in this case) allows them to see what constant term is needed to make a perfect square on the left side. They also find it to be less prone to errors.

Speaking: MLR8 Discussion Supports. To support students in producing statements about how features of the strategies are alike and how they are different, provide sentence frames for students to use when they are comparing and contrasting. For example, “_____ and _____ are alike because . . .”, “_____ and _____ are different because . . .”, “One thing that is the same is . . .”, “One thing that is different is . . . .”
Design Principle(s): Support sense-making
Action and Expression: Internalize Executive Functions. Provide students with a two-column graphic organizer to record their ideas as they compare and contrast the two solution methods. Label the left column “alike” and the right column “different.” Encourage students to use the organizer to take notes and then prepare their ideas to share with the whole class. If students are unsure how to start, tell them to number the steps in the examples and compare each step individually, placing it in the correct spot on the diagram.
Supports accessibility for: Language; Organization

Student Facing

One technique for solving quadratic equations is called completing the square. Here are two examples of how Diego and Mai completed the square to solve the same equation. 

Diego:

\(\displaystyle \begin {align} x^2+10x+9 &=0 \\x^2+10x &= \text-9 \\ x^2+10x+25 &=\text-9 + 25\\x^2+10x+25 &=16 \\ (x+5)^2 &=16\\ x+5=4 \quad & \text{or} \quad x+5=\text-4\\ x=\text-1 \quad & \text{or} \quad x=\text-9 \end{align}\)

Mai:

\(\begin {align} x^2 + 10x + 9 &= 0\\ x^2 + 10x + 9 + 16 &= 16\\ x^2+10x+25 &=16\\ (x+5)^2&=16\\ x+5=4 \quad & \text{or} \quad x+5=\text-4\\ x=\text-1 \quad & \text{or} \quad x=\text-9 \end {align}\)

Study the worked examples. Then, try solving these equations by completing the square:

  1. \(x^2+6x+8=0\)
  2. \(x^2+12x=13\)
  3. \(0=x^2-10x+21\)
  4. \(x^2-2x+3=83\)
  5. \(x^2+40=14x\)

Student Response

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

Student Facing

Are you ready for more?

Here is a diagram made of a square and two congruent rectangles. Its total area is \(x^2+35x\) square units.

A square and two congruent rectangles. Square has side length x units.
  1. What is the length of the unlabeled side of each of the two rectangles?
  2. If we add lines to make the figure a square, what is the area of the entire figure?
  3. How is the process of finding the area of the entire figure like the process of building perfect squares for expressions like \(x^2 + bx\)?

Student Response

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

Activity Synthesis

Select students to share their solutions and to display their reasoning for all to see. After each student presents, ask if others found the same solution but completed the square in a different way. Make sure students see that the steps could vary, but the solutions should be the same if equality between the two sides of the equal sign is maintained throughout the solving process.

Ask students how they could check their solutions. One way is by substituting the solutions back into the equation and seeing if the equation is true at those values of the variable. For example, to see if -4 and -2 are the right solutions to the equation \(x^2+6x+8=0\), we can evaluate \((\text-4)^2 + 6(\text-4) + 8\) and \((\text-2)^2 + 6(\text-2) + 8\) and see if each has a value of 0. Both \(16-24+8\) and \(4-12+8\) give 0, so the solutions are correct.

Lesson Synthesis

Lesson Synthesis

To close the lesson, help students connect the new method to earlier skills. Discuss questions such as:

  • “Earlier we solved \(x^2 + 10x + 9=0\) by completing the square. Could we have solved by rewriting it in factored form and using the zero product property? Why or why not?” (Yes, the equation can be rewritten as \((x+9)(x+1)=0\) and solved easily.)
  • “Is solving that equation by completing the square a quicker way?” (In this case, no.)
  • “Can \(x^2 + 10x + 9 = 20\) be solved by rewriting it in factored form?” (It can be written in factored form as \((x+9)(x+1)=20\), but that’s as far as we could go.)
  • “Can it be solved by completing the square?” (Yes. To make it a perfect square we can add 16 to each side, which makes the equation \(x^2 + 10x + 25 = 36\) or \((x+5)^2=36\), which can be solved by finding square roots of 36.)
  • “When might we prefer to complete the square than to rewrite an equation in factored form?” (Not all equations can be written in factored form so it is not always possible to solve that way. In those cases, we can solve by completing the square.)

Tell students that in upcoming lessons, we will look at other examples of equations that cannot be easily rewritten in factored form but can be solved by completing the square.

12.4: Cool-down - Make It a Perfect Square (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Turning an expression into a perfect square can be a good way to solve a quadratic equation. Suppose we wanted to solve \(x^2 - 14x +10 = \text-30\).

The expression on the left, \(x^2 - 14x +10\), is not a perfect square, but \(x^2 - 14x + 49\) is a perfect square. Let’s transform that side of the equation into a perfect square (while keeping the equality of the two sides).

  • One helpful way to start is by first moving the constant that is not a perfect square out of the way. Let’s subtract 10 from each side:​​​​​​

\(\displaystyle \begin {align} x^2 - 14x +10 - 10 &= \text-30 - 10\\ x^2 - 14x &= \text-40 \end {align}\)

  • And then add 49 to each side:​​​​​​

\(\displaystyle \begin {align} x^2 - 14x +49 &= \text-40 +49\\ x^2 - 14x+49 &= 9 \end {align}\)

  • The left side is now a perfect square because it’s equivalent to \((x-7)(x-7)\) or \((x-7)^2\). Let’s rewrite it:

\(\displaystyle (x-7)^2=9\)

  • If a number squared is 9, the number has to be 3 or -3. To finish up:​​​​​​

\(\displaystyle \begin {align} x-7=3 \quad & \text{or} \quad x-7=\text-3\\ x=10 \quad & \text{or} \quad x=4 \end{align}\)

This method of solving quadratic equations is called completing the square. In general, perfect squares in standard form look like \(x^2 + bx + \left(\frac{b}{2} \right)^2\), so to complete the square, take half of the coefficient of the linear term and square it.

In the example, half of -14 is -7, and \((\text-7)^2\) is 49. We wanted to make the left side \(x^2 - 14x + 49.\) To keep the equation true and maintain equality of the two sides of the equation, we added 49 to each side.