Lesson 12

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).

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.

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).

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.

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.

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

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.

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.

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.

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.