Lesson 9

In this activity, students find at least one solution of \(x^2-2x-35=0\) by substituting different values of \(x\), evaluating the expression, and checking if it has a value of 0. Experiencing this inefficient method puts students in a better position to appreciate why it may be desirable to write \(x^2-2x-35\) in factored form and use the zero product property.

Once students have had a chance to evaluate the expression \(x^2-2x-35\) using their chosen number for \(x\), ask if anyone found a value that made the expression equal 0. (It's 7.) Give students a couple of minutes to look for the other value that makes the expression equal 0.

If a student found that -5 makes the expression equal 0, ask them to demonstrate that \((\text-5)^2-2(\text-5)-35\) equals 0.

Discuss with students:

• "Can \(x^2-2x-35\) be written in factored form? What are the factors?" (Yes. \((x-7)\) and \((x+5)\))
• "If \(x^2-2x-35\) can be written as \((x-7)(x+5)\), can we solve \((x-7)(x+5)=0\) instead?" (Yes) "Will the solutions change if we use this equation?" (No. The equations are equivalent, so they have the same solutions.)
• "Why might someone choose to rewrite \(x^2-2x-35=0\) and solve \((x-7)(x+5)=0\) instead? (Because the expression is equal to 0, rewriting it in factored form allows us to use the zero product property to find both solutions. It may be more efficient than substituting and evaluating many values for \(x\). It also makes it possible to see how many solutions there are, which is not always easy to tell when the quadratic expression is in standard form.)

In this activity, students solve a variety of quadratic equations by integrating what they learned about rewriting quadratic expressions in factored form and their understanding of the zero product property. They begin by analyzing and explaining the steps in a solution strategy, and then applying their observations to solve other equations, both of which require sense making and perseverance (MP1). Students practice attending to precision (MP6) as they study solution steps and communicate what each step does or means.

As students work, notice the equations many students find challenging and those on which errors are commonly made. Discuss these challenges and errors during the activity synthesis.

Arrange students in groups of 2. Display the worked example in the activity statement for all to see.

Give students 1 minute of quiet time to study the example. Ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions with a partner.

Students may notice that:

• The right side is equal to 0 after the first step.
• There are two equations starting in step 3.
• The product of 11 and -9 is the constant term in step 2.

Students may wonder:

• why there are two equations in the last two steps
• where the \(\text-2n\) went when the expression on the left side was written in factored form
• whether Tyler's work is correct

Then, give students a moment to work quietly on the first question. Encourage students to use relevant mathematical vocabulary in their explanations (such as constant term, squared term, factored form, and zero product property).

Before students proceed to the second set of questions, pause for a class discussion. Ask students to explain each step of Tyler’s solving process and record their explanations for all to see. Encourage students to use reasoning language as opposed to position language: for instance, say "he subtracted 99 from both sides" rather than "he moved the 99 over to the other side."

Make sure students recognize that going from the second line to the third line in Tyler’s work involves finding two factors of -99 that add up to -2. Also discuss why there are two equations at the end. Students should recall that if two numbers multiply to equal 0, then one of the factors must be 0.

Ask students to solve as many equations in the second question as time permits.

Consider displaying the solutions for all to see and discussing only the equations that students found challenging and any common errors.

The last equation is unlike most equations students have seen. Invite students to share how they solved that equation. Discuss questions such as:

• “Can we use the zero product property to write \(x+4=0\) and \(x+5=0\)? Why or why not?” (No. The zero product property can be applied only to products that equal 0. The expression \((x+4)(x+5)-30\) has a product for one of its terms, but the expression itself is a difference.)
• “How can it be solved, other than by graphing?” (We can expand the factored expressions using the distributive property and write an equivalent equation: \(x^2 + 9x + 20 -30 =0\) or \(x^2 + 9x -10 =0\). This last equation can then be written as \((x+10)(x-1)=0\), which allows it to be solved.)

This activity reinforces what students learned earlier about the connections between the solutions of a quadratic equation and the zeros of a quadratic function. Previously, students were given equations and asked to graph them to determine the number of solutions and their values. Here, they are prompted to work the other way around: to write an equation to represent a quadratic function with only one solution. To do so, students need to make use of the structure of the factored form and their knowledge of the zero product property (MP7).

Give students access to graphing technology.

Invite students to share how they solved the equation algebraically. Next, invite students to share the equations they generated. Record and display them for all to see.

Students most likely have written equations in the form of \(g(x)=(x+m)(x+m)\). Ask students why the factored form, rather than the standard form, might have been preferred. Highlight that by using the same expression for the two factors, we know that the solution to \((x+m)(x+m)=0\) will be a single number.

Ask students to describe the graph of a quadratic function with one solution. Point out that this means that the function will have only one zero, and the graph of the function will have a single horizontal intercept.