In this activity, students use the structure they saw earlier to rewrite squared expressions of the form , where is not 1, into standard form. Through repeated reasoning, students see how the coefficient is related to the values of and when the expression is expanded into , and how the is related to (MP8).

Monitor for strategies students use to rewrite the expressions from factored form to standard form. Here are some strategies students may use from less efficient to more efficient:

• Drawing rectangular diagrams
• Applying the distributive property, writing out all the expanded terms, and combining like terms
• Generalizing that when expanding an expression such as :
  • The coefficient 4 in gets squared, so in is .
  • The coefficient 4 in gets multiplied by the constant term 1 twice, so is .
  • The constant term 1 in is squared, so is .

Identify students using each strategy and invite them to share during the class discussion.

The purpose of the discussion is to express a general pattern for the connection between an expression of the form  and the same expression in standard form.

Invite previously selected groups to share their strategies for rewriting the expressions. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display their work for all to see.

Display the general form of a quadratic expression in standard form, , and the general form of a quadratic expression from this activity in factored form, .

Connect the different responses to the learning goals by asking questions such as:

• “What is the relationship between  in standard form and  and  from the factored form? How can we see it in each expansion strategy?” ( . It is in the top left box of the diagram, the product of  with itself in the distribution method, and part of the pattern in problems of this type.)
• “What is the relationship between in standard form and and from the factored form? How can we see it in each expansion strategy?” ( . It is the sum of the top right and bottom left boxes in the diagram. When using the distributive property, the product of  and appears twice, so the sum is .)
• “What is the relationship between in standard form and and from the factored form? How can we see it in each expansion strategy?” ( . It is in the bottom right box of the diagram, the product of  with itself in the distribution method, and part of the pattern in problems of this type.)

Revisit the conjectures made during the Warm-up briefly to address whether they are supported by these patterns.

Then, discuss how students used their insights from the first question to help them identify perfect squares, or turn expressions into perfect squares, in the second question. Make sure students see the structure behind the values of , , and when the expression is a perfect square. This structure will help them complete the square in the next activity.

If students struggle to follow the generalized relationships between and its equivalent expression of the form (as discussed in the Launch), consider revisiting a concrete example from an earlier activity—for example, . Ask students to relate each term in the squared factor to the terms of its expression in standard form, .

If students get stuck finding the constant, have them write something like , then expand the right side and compare the expanded form to the left side of the equation to help them move forward.

Ask students to write the standard-form expression they invented (from the blank row of the table) on a piece of scrap paper, except without the constant term. Invite them to switch papers with a partner, and complete the square and write each other’s expressions in factored form. If they get stuck, encourage them to talk with their partner to work together and fix any mistakes.

If time permits, invite students to share their responses and strategies. Discuss questions such as:

• “How did you know what value to use for in, say, ?” ( , so is either 10 or -10. If is 10, then and . We know that , so is , or 16. If is -10,then  and  . Squaring -4 still gives 16.)
• “We might summarize the order of completing the square as this: First, find , then use that to find , then find . How do you do each of those three steps?” ( )
• “How did you know whether  would be positive or negative?” ( is some number squared, so it will always be positive.)
• “In the equation , the expression on the left already has 10 as a constant term, but it is not a perfect square. How do we make it a perfect square?” (25 would make it a perfect square, so we can either add 15 to each side of the equation or first subtract 10 from each side and then add 25.)

Note that  can be positive or negative when using this process to find , which will affect the sign of . But because ,  will always be positive, so it does not matter which signs of  and  are used to find .

As students explain their solution methods for the second question, record and display their reasoning for all to see, or display a worked solution, such as:

This activity is optional. It shows three different methods for solving an equation: by rewriting it in factored form, by transforming it into an expression in which the squared term has a coefficient 1 and then rewriting in factored form, and by completing the square. The second method involves temporarily substituting a part of the expression with another variable so that the squared term has a coefficient of 1, which makes it easier to rewrite in factored form. Students encountered this strategy in an optional activity (“Making It Simpler”) in an earlier lesson. If students did not do that optional activity, consider doing it before using this activity.

The numbers in the equations here do not lend themselves to be easily rewritten in factored form (without guessing and checking or substitution) or as perfect squares (because the leading coefficients and the constant terms are not perfect squares), but students see that these equations can be solved with these strategies. The activity further illustrates that any of these methods can be rather tedious, providing further motivation to solve using the quadratic formula, which students will learn in a future lesson.

Consider asking students who solve the same equation using different methods to compare and contrast their solution strategies. Then, invite them to reflect on the solving process:

• “Did the numbers in certain equations make it easier to solve by one method versus the others?”
• “What are the advantages of each method? What about its drawbacks?”
• “Is there a strategy you prefer or find reliable? Which one and why?”

Tell students that in upcoming lessons, they will look at a more efficient method for solving quadratic equations.