Students solve systems of equations involving quadratics and think about solutions both algebraically and graphically. As with linear equations, solutions can be identified from graphs based on where the lines cross. Unlike systems of linear equations, systems involving quadratic functions can have 0, 1, or 2 distinct solutions, in addition to the infinitely many solutions case.

Students begin the lesson by focusing on structure to identify possible solutions to a series of systems of equations (MP7). Then students combine their skills solving systems of equations using substitution and solving quadratic equations to identify all solutions to systems of equations involving at least one quadratic function. They will continue to use these skills in future lessons involving rational functions and polynomial identities.

The last activity shows students that while algebraic solving methods are useful, technology can help us identify solutions to systems of equations where the steps needed to solve by hand are less obvious. Students also consider what a possible factor of a polynomial expression written in standard form could be, which will be the focus of the following lessons leading into the Remainder Theorem.