This Warm-up prompts students to connect quadratic equations in standard form, factored form, and their solutions. Up until this point, students have solved quadratic equations with only real solutions. This idea is key for upcoming activities where students solve quadratic equations that have no real solutions but do have non-real solutions.

Select students to share how they matched equations and solutions. The important idea to discuss is that the kind of numbers that can be solutions to these equations has expanded to include non-real complex numbers.

The purpose of this activity is for students to solve quadratic equations of the sort that result from completing the square, including equations that have solutions that involve imaginary numbers. The left-hand side of each equation uses the same expression so that students can focus on what taking the square root does in each case. In this way, students are primed to make use of structure to see how imaginary numbers arise when solving quadratic equations (MP7).

The purpose of this activity is for students to solve two related quadratic equations by completing the square. The process of solving will be nearly identical, with the difference being that one will involve square roots of a negative number and the other will not. It is not necessary for students to prove a general statement about all monic quadratic equations, but it is important for students to compare and understand why some have real solutions and others have non-real solutions.

The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to describe how many solutions an equation has, and whether those solutions are real or non-real. Students have already solved two of the equations in the previous activity, and the equation they haven’t solved involves the perfect square that connects all three of the equations. Graphically, intersects the -axis twice, which corresponds to the two real solutions to . The graph intersects the -axis once, which corresponds to the single real solution to . However, the graph doesn’t intersect the -axis, which reflects the fact that the complex solutions to can’t be described with a real number line.