Solving Quadratic Equations with the Zero Product Property

10 min

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

Student Task

What values of the variables make each equation true?

  • 6+2a=06 + 2a = 0
  • 7b=07b=0
  • 7(c−5)=07(c-5)=0
  • g⋅h=0g \boldcdot h=0

Sample Response

  • -3. Sample reasoning: To have a sum of 0, the terms must be opposites. That means 2a=-62a = \text{-}6 and a=-3a = \text{-}3.
  • 0. Sample reasoning: The only number that can be multiplied by 7 to get 0 is 0 itself.
  • 5. Sample reasoning: By a similar reasoning, c−5=0c-5=0, so c=5c=5.
  • Either g=0g=0 or h=0h=0. Sample reasoning: To multiply and get 0, at least one of the factors must be 0.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’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 ‾\underline{\hspace{.5in}}’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 ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}}, 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
Standards
Building On
  • A-REI.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.
  • A-REI.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.
  • A-REI.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.
  • 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.

15 min

10 min