Unit 8, Lesson 4
Solving Quadratic Equations with the Zero Product Property
Algebra 1
4.1
Warm-up
Standards Alignment
Building On
HSA-REI.A.1
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Addressing
Building Toward
Materials
NoneActivity Narrative
This Math Talk focuses on introducing the zero product property. It encourages students to think about how to make zero from two factors and to rely on what they know about multiplying to make zero to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve quadratic equations in factored form.
To notice the key property, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneWhat values of the variables make each equation true?
Activity Synthesis
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
Highlight explanations that state that any number multiplied by 0 is 0. Then, introduce the zero product property, which states that if the product of two numbers is 0, then at least one of the numbers is 0.
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed , so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Advances: Speaking, Representing
4.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
In this activity, students solve equations of increasing complexity and do so by reasoning. They begin with linear equations, move toward a series of quadratic expressions in factored form, and end with a cubic expression in factored form. The progression prompts students to reason about the parts and structure of the expressions (MP7), rather than to memorize steps for solving without understanding, and to notice regularity through repeated reasoning (MP8).
As students discuss their reasoning with their partner, listen for those who invoke the zero product property to explain how the last four equations could be solved, and for those who notice a pattern in how the equations could be solved.
Launch
Arrange students in groups of 2.
Give students 1–2 minutes of independent work time followed by 2–3 minutes to discuss with a partner, then hold a whole-class discussion.
If needed, remind students that some equations have more than one solution. Because we want students to use reasoning and the structure of equations to develop their solutions, discourage use of graphing technology or spreadsheets in this activity.
Activity
None4.3
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
This activity enables students to apply the zero product property to solve a contextual problem and reinforces the idea of solving quadratic equations as a way to reason about quadratic functions.
Previously, students have encountered two equivalent quadratic expressions that define the same quadratic function. Here, they work to show that two quadratic expressions—one in standard form and the other in factored form—really do define the same function.
Monitor for these likely strategies, and select students who use various strategies to share in the discussion. Students may:
- Graph both equations on the same coordinate plane and show that they coincide.
- Inspect a table of values of both equations and show that the same output results for any input.
- Use the distributive property to multiply the expression in factored form to show that .
Next, they consider whether the standard form or factored form better helps them find the zeros of the function. They then use that form to find the zeros without graphing. The work here reiterates the connections between finding the zeros of a quadratic function and solving a quadratic equation where a quadratic expression that defines a function has a value of zero.
Lesson Synthesis
To help students consolidate the ideas in the lesson, discuss questions such as:
- “How does the zero product property help us find the solutions to ?” (It tells us that either or ,” and each of these equations can be solved easily.)
- “Can you explain why the solutions to are not 3 and -4?” (The zero product property works only when the product of the factors is 0. When the product is any other number, we can’t conclude that each factor is that number.)
- “The expression is equivalent to . Can we apply the zero product property to solve ?” (Only if we rewrite the expression on the left in factored form first. We can’t use the zero product property when the expression is not a product of factors.)
- “Can we solve by performing the same operation to each side of the equation?” (No, doing that doesn’t help us isolate the variable.)
Student Lesson Summary
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0. In other words, if then either or . This property is handy when an equation we want to solve states that the product of two factors is 0.
Suppose we want to solve . This equation says that the product of and is 0. For this to be true, either or , so both 0 and -9 are solutions.
Here is another equation: . The equation says the product of and is 0, so we can use the zero product property to help us find the values of . For the equation to be true, one of the factors must be 0.
- For to be true, would have to be 2.345.
- For or () to be true, would have to be , or .
The solutions are 2.345 and .
In general, when a quadratic expression in factored form is on one side of an equation and 0 is on the other side, we can use the zero product property to find its solutions.
This property is unique to 0. Given an equation like , the factors could be 2 and 3, 1 and 6, -12 and , and , or any other of the infinite number of combinations. This type of equation does not give insight into the value of or .