Cogito
Integrated Math 2 · Chapter 3 · Lesson 3
Complex Roots of Quadratics
Every quadratic finally has two roots.
12 problems · about 21 minutes · N-CN.C.7
What this lesson teaches
The student finds complex roots of quadratics and relates them to the graph.
- Every quadratic has two roots once complex numbers are allowed.
- With real coefficients, complex roots are conjugate pairs.
- Complex roots mean the parabola misses the x-axis.
Warm Up
Straightforward practice. Get the method working first.
5 problemsx² + 16 = 0. What is the positive imaginary coefficient of a root?
Answer 4
Why 4.
Why do complex roots come in conjugate pairs?
Answer The formula adds and subtracts the same imaginary amount.
Why The plus-or-minus produces the pair.
x² + 9 = 0. One root is 3i. Imaginary coefficient of the other?
Answer -3
Why Conjugate.
x² + 4 = 0. Positive imaginary coefficient of a root?
Answer 2
Why √(−4) = 2i.
x² + 2x + 5 = 0. Real part of each root?
Answer -1
Why −2 ÷ 2.
Build It Up
The same ideas with more to keep track of.
3 problemsSame equation. Positive imaginary coefficient?
Answer 2
Why 4i ÷ 2.
How many roots does every quadratic have, counting complex ones?
Answer 2
Why The degree.
Complex roots mean how many x-axis crossings?
Answer 0
Why No real roots.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsWhich statements about complex roots are true?
Answer With real coefficients they come in conjugate pairs; The parabola does not cross the x-axis
Why A parabola always has a vertex.
x² + 25 = 0. Positive imaginary coefficient of a root?
Answer 5
Why √(−25) = 5i.
The Pair: One root is 3 + 5i. What is the imaginary coefficient of the other root?
Answer -5
Why −5.
The Discriminant: x² + 2x + 5 = 0. What is the discriminant?
Answer -16
Why −16.