Lesson 14

In this warm-up, students consider perfect squares in standard form in which the leading coefficient is not 1. They use what they know about expanding \((x+m)^2\) to analyze the expansion of an expression of the form \((kx + m)^2\) and to identify an error. In explaining and correcting the error, students practice constructing logical arguments (MP3).

Display the entire task for all to see. Give students 2 minutes of quiet think time. Select students to share their responses and how they reasoned about the error in Elena’s statement.

Select 1–2 students to share their responses. Make sure students see that expanding the expression \((kx + m)^2\) involves the same structure as expanding \((x+m)^2\). Highlight the structure using a diagram and the distributive property.

When we expand \((x+3)^2\), the squared term is \(x^2\), the linear term is 2 times \((3)(x)\), which is \(6x\), and the constant term is \(3^2\). When we expand \((2x+3)^2\), the squared term is \((2x)^2\), the linear term is 2 times \((2x)(3)\), which is \(12x\), and the constant term is \(3^2\).

Students will have opportunities to transform more of such expressions later and, with practice, will better see the effects of squaring a linear expression like \(2x+3\), in which the coefficient of the variable is not 1.

In this activity, students use the structure they saw earlier to rewrite squared expressions of the form \((kx+m)^2\), where \(k\) is not 1, into standard form. Through repeated reasoning, students see how the coefficient \(k\) is related to the values of \(a\) and \(b\) when the expression is expanded into \(ax^2+bx+c\), and how the \(m\) is related to \(c\) (MP8).

As students work, monitor for the following strategies for rewriting the expressions in the first question (from factored form to standard form):

• 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 \((4x+1)^2\):
  • The coefficient 4 in \(4x+1\) gets squared, so \(a\) in \(ax^2 + bx+c\) is \(4^2\).
  • The coefficient 4 in \(4x+1\) gets multiplied by the constant term 1 twice, so \(b\) is \(2(4)(1)\).
  • The constant term 1 in \(4x+1\) is squared, so \(c\) is \(1^2\).

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

Explain to students that they will now practice squaring expressions such as \(4x+1\) and \(\frac12 x+7\), where the coefficient of the linear term is not 1, and writing them in standard form.

Consider arranging students in groups of 2 and asking them to think quietly about each question before conferring with their partner.

Select previously identified students to share their responses and strategies for the first question, starting with students who used diagrams and ending with students who generalized the pattern in the rewriting process. As students explain, display or record their reasoning for all to see.

If not mentioned in students’ explanations, highlight the generalization noted in the activity narrative.

Then, discuss how students used their insights from the first question to help them identify perfect squares, or to turn expressions into perfect squares, in the second question. Make sure students see the structure behind the values of \(a\), \(b\), and \(c\) when the expression \(ax^2+bx+c\) is a perfect square. This structure will help them complete the square in the next activity.

Earlier, students noticed that when an expression \(kx+m\) is squared and written in standard form \(ax^2+bx+c\), the value of \(b\) is \(2km\) and the value of \(c\) is \(m^2\). In this activity, they use these observations to complete expressions to make them perfect squares and then write them as squared factors. Then, they apply this skill to solve quadratic equations.

To reiterate the connections between \((kx + m)^2\) and its equivalent expression of the form \(ax^2+bx+c\), display these two expressions for all to see and ask students:

• “If the two expressions are perfect squares and are equivalent, how is \(a\) related to \(k\)?” (\(a\) is \(k^2\).)
• “How is \(c\) related to \(m\)?” (\(c\) is \(m^2\).)
• “How is \(b\) related to \(k\) and \(m\)?” (\(b\) is \(2km\).)

Tell students they will use these insights to complete some squares. Consider keeping students in groups of 2.

If students struggle to follow the generalized relationships between \((kx + m)^2\) and its equivalent expression of the form \(ax^2+bx+c\) (as discussed in the launch), consider revisiting a concrete example from an earlier activity—for example, \((5x-2)^2\). Ask students to relate each term in the squared factor to the terms of its expression in standard form, \(25x^2 - 20x + 4\).

If further scaffolding is helpful, consider using an example with an additional variable (for instance, \((3x+n)^2\) and its counterpart in standard form, \(9x^2 + 2(3)(n)x + n^2\)) before generalizing the equivalence of the two forms in entirely abstract terms.

Ask students to write the standard-form expression they invented on a piece of scrap paper, except without the constant term. (From the blank row of the table.) 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 \(c\) in, say, \(100x^2 + 80x + c\)?” (\(k^2=100\), so \(k\) is either 10 or -10. If \(k\) is 10, then \(2(10)m=80\) and \(m=4\). We know that \(c=m^2\), so \(c\) is \(4^2\) or 16. If \(k\) is -10,then \(2(\text-10) m=80\) and \(m= \text-4\). Squaring -4 still gives 16.)
• “How did you know whether \(c\) would be positive or negative?” (\(c\) is some number squared, so it will always be positive.)
• “In the equation \(36x^2 - 60x + 10 = \text-6\), 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.)

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

\(\begin{align} 36x^2 - 60x + 10 &= \text-6\\ 36x^2 - 60x + 10 + 15 &= \text-6 + 15\\ 36x^2 - 60x + 25 &= 9\\ (6x-5)^2 &=9 \end{align}\)\( \)

\(\begin {align} 6x-5 =3 \quad &\text{or} \quad 6x-5 =\text-3\\ 6x = 8 \quad &\text{or} \quad 6x = 2\\ x = \frac86 =\frac43 \quad & \text{or} \quad x = \frac13 \end{align}\)

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 first 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 to be easily rewritten into perfect squares (because the leading coefficient 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.

Display the perfect squares students saw in the preceding activity. Ask them what they notice about the coefficient of the square term in each expression.

\(\displaystyle 100x^2 + 80x + 16\)

\(\displaystyle 36x^2 - 60x + 25\)

\(\displaystyle 25x^2 + 40x + 16\)

\(\displaystyle 0.25x^2- 14x + 196\)

Students are likely to notice that the coefficient of every \(x^2\) is a perfect square, which made it easier to rewrite the completed square as squared factors. Solicit some ideas about whether we could still complete the square if a quadratic expression does not have a perfect square for the coefficient of \(x^2\). Then, tell students that they will explore a few methods for solving such equations.

Arrange students in groups of 2. Ask one partner to study the first two methods and the other partner to study the third method, and then take turns explaining their understanding to each other. Discuss the methods, especially the third one, before students begin using them to solve equations.

Make sure students see that, in both the second and third methods, the first step involves multiplying both sides of the equation by 3 to make the coefficient of \(x^2\) a perfect square. Each term on the left side of the equation changes by a factor of 3, but the right side of the equation remains 0 because multiplying 0 by any number results in 0.

If time is limited, consider asking students to solve only three equations, using each method once.

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.