Lesson 10

Rewriting Quadratic Expressions in Factored Form (Part 4)

10.1: Which One Doesn’t Belong: Quadratic Expressions (5 minutes)

Warm-up

This warm-up prompts students to carefully analyze and compare quadratic expressions. In making comparisons, students need to look for common structure and have a reason to use language precisely (MP7, MP6). The activity also enables the teacher to hear the terminology students know and how they talk about characteristics of the different forms of expressions.

As students discuss in groups, listen for rationales that are based on the structure of the expressions. Select those students or groups to share their thinking during class discussion.

Launch

Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning about why a particular item does not belong, and together find at least one reason each item doesn't belong.

Student Facing

Which one doesn’t belong?

A. \((x+4)(x-3)\)

B. \(3x^2-8x+5\)

C. \(x^2-25\)

D. \(x^2+2x+3\)

Student Response

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

Activity Synthesis

Ask each group to share one reason why a particular expression doesn’t belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as standard form, factored form, linear term, or coefficient. Also, press students on unsubstantiated claims. For example, if they claim that B is the only one that cannot be written in factored form, ask them to show how they know.

10.2: A Little More Advanced (15 minutes)

Activity

Most of the factored expressions students saw were of the form \((x+m)(x+n)\) or \(x(x+m)\). In this activity, students work with expressions of the form \((px+m)(qx+n)\). They expand expressions such as \((2x+1)(x+10)\) into standard form and look for structure that would allow them to go in reverse (MP7), that is, to transform expressions of the form \(ax^2+bx+c\), where \(a\) is not 1 (such as \(2x^2+12x+10\)) into factored form.

Going from factored form to standard form is fairly straightforward given students’ experience with the distributive property. Going in reverse, however, is a bit more challenging when the coefficient of \(x^2\) is not 1. With some guessing and checking, students should be able to find the factored form of the expressions in the second question, but they should also notice that this process is not straightforward.

Launch

Remind students that they have seen quadratic expressions such as \(16t^2 + 800t + 400\) and \(5x^2 +21x -20\), where the coefficient of the squared term is not 1. Solicit some ideas from students on how to write the factored form for expressions such as these.

Arrange students in groups of 2–3 and ask them to split up the work for completing the first table, with each group member rewriting one expression into standard form.

Display the incomplete table in the first question for all to see, and then invite students to share the expanded expressions in standard form. Record the expressions in the right column, and ask students to make observations about them.

If not mentioned by students, point out that each pair of factors start with \(3x\) and \(x\), which multiply to make \(3x^2\). Each pair of factors also has constant terms that multiply to make 4. The resulting expressions in standard form are all different, however, because using different pairs of factors of 4 and arranging them in different orders produce different expanded expressions.

Ask students to keep these observations in mind as they complete the second question.

If time is limited, ask each group member to choose at least two expressions in the second table and rewrite them into factored form.

Representation: Internalize Comprehension. Activate or supply background knowledge. Demonstrate how students can continue to use diagrams to rewrite expressions in which the coefficient of the squared term is not 1. Invite students to begin by generating a list of factors and to test them using the diagram. Encourage students to persist with this method, reiterating the fact that they are not necessarily expected to immediately recognize which factors will work without testing them. Allow students to use calculators to ensure inclusive participation.
Supports accessibility for: Visual-spatial processing; Organization

Student Facing

Each row in each table has a pair of equivalent expressions. Complete the tables. If you get stuck, try drawing a diagram.

  1. factored form standard form
    \((3x+1)(x+4)\)  
    \((3x+2)(x+2)\)  
    \((3x+4)(x+1)\)  
  2. factored form standard form
      \(5x^2+21x+4\)
      \(3x^2+15x+12\)
      \(6x^2+19x+10\)

 

Student Response

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

Student Facing

Are you ready for more?

Here are three quadratic equations, each with two solutions. Find both solutions to each equation, using the zero product property somewhere along the way. Show each step in your reasoning.

\(x^2=6x\)

\(x(x+4)=x+4\)

\(2x(x-1)+3x-3=0\)

Student Response

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

Anticipated Misconceptions

Some students may not think to check their answers to the second question and stop as soon as they think of a pair of factors that give the correct squared term and constant term. Encourage them to check their answers with a partner by giving time to do so. Consider providing a non-permanent writing surface or extra paper so students could try out their guesses and check their work without worrying about having to erase if they make a mistake the first time or two.

Activity Synthesis

Display for all to see the incomplete table in the second question. Select students to complete the missing expressions in standard form and to briefly explain their strategy. To rewrite \(ax^2+bx+c\), students are likely to have tried putting different factors of \(a\) and of \(c\) in the factored expression such that when the factors are expanded, they yield a linear term with the coefficient \(b\).

Then, help students to reason about the factors more generally. Discuss questions such as:

  • “To rewrite expressions such as \(x^2+bx+c\), we looked for two numbers that multiply to make \(c\) and add up to \(b\). The expressions here are of the form \(ax^2+bx+c\). Are we still looking for two numbers that multiply to make \(c\)? Why or why not?” (Yes. The constant terms in the factored expression must multiply to make \(c\).)
  • “Do we need to look for factors of \(a\)? Why or why not?” (Yes. Those factors will be the coefficient of \(x\) in the factored expressions. They must multiply to make \(a\).)
  • “Are we still looking for two factors of \(c\) that add up to \(b\)? Why or why not?” (No. The value of \(b\) is no longer just the sum of the two factors of \(c\) because the two factors of \(a\) are now involved.) “How does this affect the rewriting process?” (It makes it more complicated, because now there are four numbers to contend with, and there are many more possibilities to consider.)

Tell students that we’ll investigate a bit further quadratic equations in the form of \(ax^2+bx+c\) where \(a\) is not 1, and see if there are manageable ways to rewrite such equations in factored form so that they can be solved.

Speaking, Representing: MLR8 Discussion Supports. Give students time to make sure that everyone in the group can explain or justify each step or part of the problem.  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. Then 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

10.3: Timing A Blob of Water (15 minutes)

Activity

Earlier, students were given quadratic expressions of the form \(ax^2+bx+c\) where the \(a\) is not 1. They found that the rewriting process was a bit more involved but was not impossible, at least for the problems at hand.

In this activity, they encounter a real-world example in which they struggle to find a combination of rational factors of \(a\) and \(c\) that would produce the value of \(b\). (Realistic quadratic functions don’t always have rational numbers for their zeros, so the quadratic expressions that define them cannot always be written in factored form. At this point, students are not yet considering rational and irrational solutions and are not expected to know why some expressions cannot be easily written in factored form.)

By graphing the function, students discover that the horizontal intercepts are decimals (rounded to different decimal places, depending on the graphing technology used). The graph allows them to estimate the solution, but that is as far as they could go. The challenges in this activity set the stage for introducing a more productive technique for solving quadratic equations.

Launch

Keep students in groups of 2. Let students become briefly frustrated by their unsuccessful attempts to find factors of the expression in standard form, but move them on to the last question after a few minutes. Provide access to devices that can run Desmos or other graphing technology. 

Reading: MLR6 Three Reads. Use this routine to support reading comprehension, without solving, for students. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (An engineer is designing a water fountain). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for, and amplify, the important quantities that vary in relation to each other in this situation: height of a drop of water, in meters, and time, in seconds. After the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem.
Design Principle: Support sense-making
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure student access to written directions, word problems and other text-based content. Clarify any unfamiliar terms or phrases.
Supports accessibility for: Language; Conceptual processing

Student Facing

An engineer is designing a fountain that shoots out drops of water. The nozzle from which the water is launched is 3 meters above the ground. It shoots out a drop of water at a vertical velocity of 9 meters per second.

Function \(h\) models the height in meters, \(h\), of a drop of water \(t\) seconds after it is shot out from the nozzle. The function is defined by the equation \(h(t)=\text-5t^2+9t+3\).

How many seconds until the drop of water hits the ground?

  1. Write an equation that we could solve to answer the question.
  2. Try to solve the equation by writing the expression in factored form and using the zero product property.
  3. Try to solve the equation by graphing the function using graphing technology. Explain how you found the solution.

Student Response

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

Activity Synthesis

Ask students to share some challenges they came across when trying to rewrite the expressions in factored form. Solicit some ideas about why this equation presented those challenges. Then, discuss how they found or estimated the solution by graphing.

The approximate solution to the equation, given by the zero of the function and the \(x\)-intercept of the graph, is 2.087 seconds. Some graphing tools would give an approximation with a longer decimal expansion, giving a clue that it might be trickier to rewrite the equation in factored form. After all, when finding factors, we usually look for integers. (Some quadratic expressions containing non-integer rational numbers can still be written in factored form. For example, \(x^2 + \frac 34 + \frac18\) can be written as \((x+\frac12)(x+\frac14)\).)

Highlight that some equations are difficult or impossible to rewrite in factored form. In fact, when quadratic models appear in real life, this is usually the case. Graphing is a way to solve these equations, but there are other techniques, which students will learn over the next several lessons.

10.4: Making It Simpler (25 minutes)

Optional activity

This activity is optional. It allows students to investigate another way (beside guessing and checking) for finding the factors of quadratic expressions in standard form where the leading coefficient is not 1. The method involves temporarily substituting the squared term with another variable so that the leading coefficient is 1 (which makes it easier to spot the factors of the expression), and then substituting the original variable back once the expression is in factored form.

Students may find the reasoning and substitution processes challenging. Transitioning from, for example, \(9x^2+21x+10\) to \((3x)^2+7(3x)+10\), and then to \(N^2+7N+10\) requires abstract reasoning. When the leading coefficient is not a square number (as in the second half of the activity), an additional step of multiplying is needed to make the leading coefficient a square number. To keep the value of an expression unchanged, students need to remember that they can only multiply the expression by 1, but the 1 can be obtained by, say, multiplying by 5 and then by \(\frac15\).

Before offering support, allow students to make sense of these processes and to discuss with their partner. The work encourages students to persevere in sense making and problem solving (MP1) and to make use of structure (MP7). It also prompts students to attend to precision (MP6). As they make symbolic substitutions, students need to be clear about what the variables stand for and how the substitutions transform the expressions.

Later in the unit, students will encounter the use of a placeholder, such as done here, as a way to derive the quadratic formula. If desired, this activity can be done at that point to warm students up to the idea of using a placeholder.

If time is limited and if desired, the activity could be split into two halves and done separately (over two class periods).

Launch

Display these expressions for all to see and ask students which expressions would be easier to write in factored form and why.

\(4x^2 - 12x + 8\)

\(t^2 - 6t + 8\)

\(d^2 + d - 20\)

\(9x^2 + 3x - 20\)

Students are likely to say that the second and the third expressions are easier because the coefficient of the squared term in each of those is 1 (or there isn’t another number that needs to be factored aside from the constant term). Tell students that they will study another strategy that can simplify the process of rewriting quadratic expressions into factored form.

Consider arranging students in groups of 2 and asking them to think quietly about at least the first couple of problems before discussing with their partner. After students have had a chance to make sense of the first worked example, pause for a class discussion. Make sure that students can follow what is happening in the shown steps before trying to apply it with new expressions.

Pause for another class discussion after students have analyzed the second worked example. Before students proceed to the last question, clarify what is happening in each step of the rewriting process when the leading coefficient is not a square number.

Student Facing

Here is a clever way to think about quadratic expressions that would make it easier to rewrite them in factored form.

\(9x^2+21x+10 \\\\ (3x)^2+7(3x)+10 \\\\ N^2+7N+10\\\\ (N+2)(N+5) \\\\(3x+2)(3x+5)\)

  1. Use the distributive property to expand \((3x+2)(3x+5)\). Show your reasoning and write the resulting expression in standard form. Is it equivalent to \(9x^2+21x+10\)?
  2. Study the method and make sense of what was done in each step. Make a note of your thinking and be prepared to explain it.
  3. Try the method to write each of these expressions in factored form.

    \(4x^2+28x+45\)

    \(25x^2-35x+6\)

  4. You have probably noticed that the coefficient of the squared term in all of the previous examples is a perfect square. What if that coefficient is not a perfect square?

    Here is an example of an expression whose squared term has a coefficient that is not a squared term.

    \(5x^2+17x+6 \\\\ \frac15 \boldcdot 5 \boldcdot (5x^2 + 17x + 6)\\\\ \frac15 (25x^2 + 85x + 30) \\\\ \frac15 ((5x)^2 + 17 (5x) + 30)\\\\ \frac15 (N^2 + 17N + 30)\\\\ \frac15 (N+15)(N+2) \\\\ \frac15 (5x+15)(5x+2) \\\\(x+3)(5x+2)\)

    Use the distributive property to expand \((x+3)(5x+2)\). Show your reasoning and write the resulting expression in standard form. Is it equivalent to \(5x^2+17x+6\)?

  5. Study the method and make sense of what was done in each step and why. Make a note of your thinking and be prepared to explain it.
  6. Try the method to write each of these expressions in factored form.

    \(3x^2+16x+5\)

    \(10x^2-41x+4\)

Student Response

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

Anticipated Misconceptions

In the final steps of the last question, students multiply a number by a pair of factors, for example, \(\frac13 (3x+15)(3x+1)\). Some students may mistakenly apply the distributive property and multiply \(\frac13\) to both \((3x+15)\) and \((3x+1)\). Remind students that the distributive property governs multiplication over addition and subtraction, and that \((3x+15)\) and \((3x+1)\) are being multiplied together, not added or subtracted.

Activity Synthesis

Invite students to share their attempts to rewrite the expressions using the method they just learned. Discuss questions such as:

  • “Do you think this method is simpler or harder than guessing and checking? Is it quicker or slower?”
  • “In what ways is this method simpler? In what ways is it harder?”
Speaking, Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. After each student shares, provide the class with the following sentence frames to help them clarify their understanding of the speaker’s ideas and to press for further details: “How did you get . . . ?”, “Why did you . . . ?”, “How do you know . . . ?” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. 
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

Emphasize the point that rewriting quadratic expressions in factored form is not always an efficient or effective way to find solutions to quadratic equations. If time permits, consider generalizing the relationship between \(ax^2+bx+c\) and its factored form to help students see why it can be a rather tricky process:

  • We need to find two factors of \(a\). Let’s call them \(p\) and \(q\).
  • We need to find two factors of \(c\). Let’s call them \(m\) and \(n\).
  • The factored form can be written as \((px + m)(qx +n)\).
  • To see what \(p\), \(q\), \(m\), and \(n\) should be, we essentially have to apply the distributive property to the factors to get: \((px)(qx) + (np)x + (mq)x + mn\) or \((pq)x^2 + (np+mq)x + mn\).
  • We see that \(pq\) must multiply to make \(a\) and \(mn\) must multiply to make \(c\).
  • The challenge now is to find the right combination of \(p\) and \(q\) (factors of \(a\)) and \(m\) and \(n\) (factors of \(c\)) such that \(np + mq\) is equal to \(b\).

Highlight that for some quadratic expressions, the right pairs of factors might be easily spotted, but for others, the process of guessing and checking can get pretty cumbersome, especially if \(a\) and \(c\) have many pairs of factors. There is also no guarantee that we will find a combination that works because some expressions do not have rational solutions (so we cannot find factors with rational numbers). This means that if we rely on putting an expression in factored form to solve an equation, we may get stuck. If we rely on graphing, the solutions may not be exact. We need another way!

Tell students that in upcoming lessons we will look at other techniques that allow us to solve without rewriting expressions in factored form.

10.5: Cool-down - How Would You Solve This Equation? (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Only some quadratic equations in the form of \(ax^2 + bx +c=0\) can be solved by rewriting the quadratic expression into factored form and using the zero product property. In some cases, finding the right factors of the quadratic expression is quite difficult.

For example, what is the factored form of \(6x^2+11x-35\)?

We know that it could be \((3x + \boxed{\phantom{30}})(2x+\boxed{\phantom{30}})\), or \((6x+\boxed{\phantom{30}})(x+\boxed{\phantom{30}})\), but will the second number in each factor be -5 and 7, 5 and -7, 35 and -1, or -35 and 1? And in which order?

We have to do some guessing and checking before finding the equivalent expression that would allow us to solve the equation \(6x^2+11x-35=0\).

Once we find the right factors, we can proceed to solving using the zero product property, as shown here:​​​​​​

\(\displaystyle \begin {align} 6x^2+11x-35&=0\\ (3x-5)(2x+7)&=0\\ \end {align}\)

\(\displaystyle \begin {align} 3x-5=0 \quad &\text{or} \quad 2x+7=0\\ n=\frac53 \quad &\text{or} \quad x= \text- \frac72\ \end {align}\)

What is even trickier is that most quadratic expressions can’t be written in factored form!

Let’s take \(x^2-4x-3\) for example. Can you find two numbers that multiply to make -3 and add up to -4? Nope! At least not easy-to-find rational numbers.

We can graph the function defined by \(x^2-4x-3\) using technology, which reveals two \(x\)-intercepts, at around \((\text-0.646,0)\) and \((4.646,0)\). These give the approximate zeros of the function, -0.646 and 4.646, so they are also approximate solutions to \(x^2-4x-3=0\).

Parabola facing up with 2, x intercepts at - 0 point 4,6,4 comma 0 and at 6 point 4,6,4 comma 0.

The fact that the zeros of this function don’t seem to be simple rational numbers is a clue that it may not be possible to easily rewrite the expression in factored form.​​​​​​

It turns out that rewriting quadratic expressions in factored form and using the zero product property is a very limited tool for solving quadratic equations.

In the next several lessons, we will learn some ways to solve quadratic equations that work for any equation.