Unit 8, Lesson 9
Solving Quadratic Equations by Using Factored Form
Algebra 1
9.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students find at least one solution of by substituting different values of , evaluating the expression, and checking if it has a value of zero. Experiencing this inefficient method puts students in a better position to appreciate why it may be desirable to write in factored form and use the zero product property.
Launch
Once students have had a chance to evaluate the expression , using their chosen number for , 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.
Activity Synthesis
If a student found that -5 makes the expression equal 0, ask them to demonstrate that equals 0.
Discuss with students:
- "Can be written in factored form? What are the factors?" (Yes, and .)
- "If can be written as , can we solve 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 and solve 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 . 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.)
9.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
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, look for the equations many students find challenging and those on which errors are commonly made. Discuss these challenges and errors during the Activity Synthesis.
Launch
Arrange students in groups of 2. Give students quiet work time on the first question. Encourage students to use relevant mathematical vocabulary, such as “constant term,” “squared term,” “factored form,” and “zero product property,” in their explanations .
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 each side" 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.
Representation: Internalize Comprehension. Begin by asking, “Do these problems remind anyone of something we have done before?”
Supports accessibility for: Memory, Attention
Supports accessibility for: Memory, Attention
9.3
Activity
Instructional Routines
Graph ItMaterials
To Gather
Graphing technologyActivity Narrative
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 the values of the solutions. 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).
Launch
Give students access to graphing technology.
Activity
NoneLesson Synthesis
Display a series of equations that, prior to this lesson, students could solve only by graphing. For instance:
Ask students to choose an equation that they think they could solve without graphing. Then, ask them to explain to a partner why they believe they could solve the equation.
Consider displaying for all to see and using it as an example: “I think I can solve because I know the expression on the left is equivalent to and I can rewrite the equation as , which I can solve by rearranging the terms.”
Then, ask if any of the equations appear to be unsolvable other than by graphing and why. Of the equations shown here, is the only one that students aren’t yet equipped to solve because the coefficient of is not 1. Students will begin looking at such equations in an upcoming lesson.
Student Lesson Summary
Recently, we learned strategies for transforming expressions from standard form to factored form. In earlier lessons, we have also seen that when a quadratic expression is in factored form, we can find values of the variable that make the expression equal zero. Suppose we are solving the equation , which says that the product of and is 0. By the zero product property, we know this means that either or , which then tells us that 0 and -4 are solutions.
Together, these two skills—writing quadratic expressions in factored form and using the zero product property when a factored expression equals 0—allow us to solve quadratic equations given in other forms. Here is an example:
When a quadratic equation is written as as an expression in factored form equal to 0, we can also see the number of solutions the equation has.
In the previous example, it is not obvious how many solutions there are when the equation is in the form . When the equation is rewritten as , we can see that there are two numbers that could make the expression equal 0: 14 and -10.
How many solutions does the equation have?
Let’s rewrite it in factored form: . The two factors are identical, which means that there is only one value of that makes the expression equal 0. The equation has only one solution: 10.