Lesson 17
Completing the Square and Complex Solutions
Lesson Narrative
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).
Learning Goals
Teacher Facing
 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.
Student Facing
 Let’s find complex solutions to quadratic equations by completing the square.
Required Materials
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.)
Learning Targets
Student Facing
 I can find complex solutions to quadratic equations by completing the square.
CCSS Standards
Print Formatted Materials
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Additional Resources
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