In earlier courses, students developed strategies for solving quadratic equations. Earlier in this unit, students developed the concept of complex numbers. In this lesson, students connect these ideas by solving quadratic equations whose solutions are nonreal complex numbers. Students complete the square to analyze the conditions that lead quadratic equations with real coefficients to have 1 real solution, 2 real solutions, or 2 nonreal solutions.
Students are making use of structure when they complete the square with \(x^2+bx+c=0\) to understand the relationship between the constant coefficient, the coefficient of \(x\), the number of solutions, and the type of solutions (MP7).
Lesson overview
 17.1 Warmup: Creating Quadratic Equations (5 minutes)
 17.2 Activity: Sometimes the Solutions Aren't Real Numbers (10 minutes)

17.3 Activity: Finding Complex Solutions (15 minutes)
 Includes "Are you Ready for More?" extension problem
 17.4 Optional Activity: Can You See the Solutions on a Graph? (10 minutes)
 Lesson Synthesis
 17.5 Cooldown: Make One (5 minutes)
Learning goals:
 Calculate complex solutions to quadratic equations by completing the square.
 Compare and contrast quadratic equations with real and with nonreal complex solutions.
 Explain how to solve equations that have nonreal complex solutions.
Learning goals (student facing):
 Let’s find complex solutions to quadratic equations by completing the square.
Learning targets (student facing):
 I can find complex solutions to quadratic equations by completing the square.
Required materials:
 Graphing technology
Required preparation:
 Acquire devices that can run Desmos (recommended) or other graphing technology.
 It is ideal if each student has their own device. (Desmos is available under Math Tools.)
Standards:
 This lesson builds towards the standards: CCSS.HSAREI.B.4.bMS.AREI.4bCCSS.HSNCN.C.7MS.NCN.7MO.A1.REI.A.2cMO.A2.REI.A.1
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