Complex Numbers as Solutions to Equations

Complex Numbers as Solutions to Equations

Student Outcomes

• Students solve quadratic equations with real coefficients that have complex solutions. They recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.

New York State Common Core Math Algebra II, Module 1, Lesson 38

Worksheets for Algebra 2

Classwork

Opening Exercise

1. The expression under the radical in the quadratic formula, 𝑏² − 4𝑎𝑐, is called the discriminant. Use the quadratic formula to solve the following quadratic equations. Calculate the discriminant for each equation.
   a. 𝑥² − 9 = 0
   b. 𝑥² − 6𝑥 + 9 = 0
   c. 𝑥² + 9 = 0
2. How does the value of the discriminant for each equation relate the number of solutions you found?

Example 1
Consider the equation 3𝑥 + 𝑥² = −7.
What does the value of the discriminant tell us about number of solutions to this equation?

Solve the equation. Does the number of solutions match the information provided by the discriminant? Explain.

Exercise
Compute the value of the discriminant of the quadratic equation in each part. Use the value of the discriminant to predict the number and type of solutions. Find all real and complex solutions.
a. 𝑥² + 2𝑥 + 1 = 0.
b. 𝑥² + 4 = 0
c. 9𝑥² − 4𝑥 − 14 = 0
d. 3𝑥² + 4𝑥 + 2 = 0
e. 𝑥 = 2𝑥² + 5
f. 8𝑥² +4𝑥 + 32 = 0

Lesson Summary

• A quadratic equation with real coefficients may have real or complex solutions.
• Given a quadratic equation 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0, the discriminant 𝑏² − 4𝑎𝑐 indicates whether the equation has two distinct real solutions, one real solution, or two complex solutions.

If 𝑏² − 4𝑎𝑐 > 0, there are two real solutions to 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0.
If 𝑏² − 4𝑎𝑐 = 0, there is one real solution to 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0.
If 𝑏² − 4𝑎𝑐 < 0, there are two complex solutions to 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0.

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