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).

Select students to explain how they got their solutions.

If not brought up by students, ask students to discuss what is the same and different between the three problems and how these similarities and differences affect the solutions.

• For , there is only one solution because the only number that squares to make 0 is 0.
• For , there are two real solutions because there are two real numbers that square to make 1.
• For , there are two non-real solutions because there are two imaginary numbers that square to make -1, and then adding 5 to each side results in complex solutions.

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.

Select students to share their solutions. Record and display their thinking for all to see. If not brought up during the discussion, ask students to clarify why one equation has real solutions while the other has complex solutions.

Display these three equations for all to see:

Ask students, “How many solutions do these equations have? Which, if any, of these equations has solutions that involve imaginary numbers?” ( has two real solutions, has one real solution, and has two non-real solutions.)

Compare each left-hand side to the perfect square , which is equal to . When completing the square, the first equation is equivalent to , the second equation is equivalent to , and the third equation is equivalent to .

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.

If students attempt to plot the complex solution in the coordinate plane as if it were the complex plane, consider saying:

• “Explain your graph to me.”

• “What is the same and what is different about the points and ?”

Select students to share their solutions and connections between the graphs and the equations. Discuss the idea that if a graph of a quadratic function doesn’t cross a given horizontal line (in this case, ) this doesn’t indicate that the related equation has no solutions, but rather that it has no real solutions that can be graphed with real number lines.