Lesson 17

This warm-up prompts students to think about substituting non-real complex numbers for variables in quadratic equations for the first time. Up until this point, students have only substituted real numbers for variables. This opens up the possibilities for what numbers could be solutions to equations. 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 \((x-5)^2\) 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).

Arrange students in groups of 2. Give a few minutes of quiet work time before asking students to share their reasoning with their partner. If there is disagreement, encourage students to work to reach agreement. Follow with whole-class discussion.

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 \((x-5)^2=0\), there is only one solution because the only number that squares to make 0 is 0. For \((x - 5)^2 = 1\), there are two real solutions because there are two real numbers that square to make 1. For \((x - 5)^2 = \text-1\), 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 non-real, 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:

• \(x^2+14x+44=0\)
• \(x^2+14x+49=0\)
• \(x^2+14x+54=0\)

Ask students, “How many solutions do these equations have? Which, if any, of these equations has solutions that involve imaginary numbers?” (\(x^2+14x+44\) has two real solutions, \(x^2+14x+49\) has one real solution, and \(x^2+14x+54\) has two non-real solutions. Compare each left hand side to the perfect square \((x+7)^2\), which is equal to \(x^2+14x+49\). When completing the square, the first equation will have \((x+7)^2 = 5\), the second equation will have \((x+7)^2 = 0\), and the third equation will have \((x+7)^2 = \text-5\).)

This activity is optional because it goes beyond the depth of understanding required by the standards.

The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to tell 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, \(y=f(x)\) intersects the \(x\)-axis twice, which corresponds to the two real solutions to \(x^2 - 8x + 13 = 0\). The graph \(y=g(x)\) intersects the \(x\)-axis once, which corresponds to the single real solution to \(x^2 - 8x + 16 = 0\). However, the graph \(y=h(x)\) doesn’t intersect the \(x\)-axis, which reflects the fact that the complex solutions to \(x^2 - 8x + 19 = 0\) can’t be described with a real number line.

Provide access to graphing technology.

Some students may attempt to plot the complex solutions in the coordinate plane as if it were the complex plane. Remind these students that points in the coordinate plane are pairs of real numbers \((x,y)\), whereas points in the complex plane are single complex numbers. In other words, the point \((2,4)\) in the coordinate plane is a different kind of object than the point \(2+4i\) in the complex plane.

Select students to share their solutions and discuss the 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, \(x=0\)) 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.