Students build on what they know about solving systems of linear equations and make use of structure (MP7) to solve systems of equations with at least one quadratic equation. In this activity, students focus on finding all points of intersection algebraically, so graphing technology is not an appropriate tool.

The purpose of this discussion is for students to see and understand the different ways their classmates solved the questions in the activity. Select one student per question to share their solutions. Encourage students to ask clarifying questions about why different solving steps were used. After each student shares, ask if any students solved the question a different way, and invite those students to share their steps.

If not brought up in students’ explanations for the 2nd or 3rd questions, display this set of equations for all to see:

Ask students:

• “What is happening in this step?” (Both sides of the equations are divided by .)
• “How is this different from other examples we have seen?” (In the example from the Launch, a linear term was subtracted from both sides of the equation instead of being divided.)
• “Will this division strategy work?” (Using this division strategy gives one value of (-6), but there is another solution that is missing.)
• “What would happen in this strategy if was 4?” (This strategy wouldn’t make sense because then we would be dividing by 0.)

Conclude the discussion by asking students if it is possible for a system of equations with two quadratic functions to have 3 solutions. If possible, display a graph with two quadratics whose shapes can be manipulated, to help convince students that 2 is the maximum number of solutions for distinct quadratic functions.

Students use graphing technology to solve a system. The graphing technology allows students to identify solutions to equations that can be tedious to do by hand. 

The question about possible factors is meant to preview the work in upcoming lessons in which students will identify factors of polynomial expressions written in standard form and eventually show that for a polynomial , if , then is a factor of .

If students are unsure of how to find a possible factor of  , consider asking:

• “What do you know about the factors of a polynomial?”
• "How does the value of  that you calculated using the -value where the functions intersect connect with the idea of a polynomial's factors?"

The goal of this discussion is for students to see how to use technology to answer an algebraic question involving a higher-degree polynomial. Invite students to share the polynomial that they wrote. If possible, display several versions of for all to see, and use the graph to estimate the -values where . If this is not possible, ask students to share the solutions they estimated. Students should understand that not all polynomials can be factored by hand, and that technology is a valuable tool for helping us interpret such polynomials.

Then ask students to suggest possible factors of . After a few suggestions, ask students how they could prove something is a factor. For now, it is important that students understand that the only way to prove something like is a factor of is to find the quadratic function that satisfies  , where , , and are real numbers. Tell students that the next lessons are about how to do this.