Curriculum  /  Math  /  11th Grade  /  Unit 2: Quadratics  /  Lesson 7

Quadratics

Lesson 7

Math

Unit 2

11th Grade

Lesson 7 of 11

Objective

Identify solutions that are non-real from a graph and an equation using the discriminant. Define imaginary and complex numbers.

Common Core Standards

Core Standards

  • N.CN.A.1 — Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
  • N.CN.C.7 — Solve quadratic equations with real coefficients that have complex solutions.
  • A.REI.B.4.B — Solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Criteria for Success

  1. Describe the discriminant as the part of the quadratic formula that helps to determine whether a quadratic function has real, complex, or a double root.
  2. Use the discriminant to name the number and kind of roots of quadratic function.
  3. Describe how complex roots look in a quadratic function on a coordinate plane, and sketch a graph of a quadratic function with complex roots. 
  4. Distinguish an imaginary number, a number that when squared forms a negative number, from a complex number, a number with an imaginary part and a real part. 
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Anchor Problems

25-30 minutes

Problem 1

Below are the vertex form of a quadratic equation, the graph of a quadratic equation, and the standard form of a quadratic equation. All three representations are of the same function.

$${f(x)=2(x-1)^2+3}$$ $${f(x)=2x^2-4x+5}$$

Describe how you know, using only the representation in each column, that the function has non-real roots.

Guiding Questions

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Problem 2

Below is an equation that represents the definition of an “imaginary” number.

$${i = \sqrt{-1}}$$

  1. Write the following expression using an imaginary number. 
     

$${3\sqrt{-4}}$$

  1. Write the following expression without an imaginary number. 

$${5i^2}$$

Guiding Questions

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Target Task

5-10 minutes

Problem 1

Write the following as an imaginary number.

$${\sqrt{-72}}$$

Why is this not a complex number?

Problem 2

What kind of roots will the following quadratic function have?

$${4x-3x^2=10}$$

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems that show the definition of imaginary numbers and how to write imaginary numbers and complex numbers appropriately. Example: “Write the following number as an imaginary number $${(\sqrt{-3})^2}$$."
  • Include problems such as “Write the number 81 as a product of imaginary numbers. Why do you need four imaginary numbers to get a positive number?”

Next

Add, subtract, and multiply complex numbers.

Lesson 8
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Features of Quadratic Functions

Topic B: Imaginary Solutions and Operating with Complex Numbers

Topic C: Applications, Systems, and Inverse with Quadratics

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