# 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 non-real 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 non-real 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 non-real complex solutions.
- Explain how to solve equations that have non-real 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.