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Cogito

Integrated Math 2 · Chapter 3 · Lesson 3

Practice Worksheet

Complex Roots of Quadratics

Every quadratic finally has two roots.

12 problems · about 21 minutes · N-CN.C.7

NameDate

Figure — use these to answer the problems

Lift the parabola clear of the axis. It has no real roots now, but it still has two complex ones.
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y = 1x² + 2x + 5
a1
b2
c5
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. x² + 16 = 0. What is the positive imaginary coefficient of a root?

    Answer

  2. Why do complex roots come in conjugate pairs?

    • The formula adds and subtracts the same imaginary amount.
    • It happens to work out that way.

  3. x² + 9 = 0. One root is 3i. Imaginary coefficient of the other?

    Answer

  4. x² + 4 = 0. Positive imaginary coefficient of a root?

    Answer

  5. x² + 2x + 5 = 0. Real part of each root?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. Same equation. Positive imaginary coefficient?

    Answer

  2. How many roots does every quadratic have, counting complex ones?

    Answer

  3. Complex roots mean how many x-axis crossings?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Which statements about complex roots are true?

    • With real coefficients they come in conjugate pairs
    • The parabola does not cross the x-axis
    • The parabola has no vertex
    • The quadratic has only one root
    • Tick every box that applies.

  2. x² + 25 = 0. Positive imaginary coefficient of a root?

    Answer

  3. The Pair

    One root is 3 + 5i. What is the imaginary coefficient of the other root?

    Answer

  4. The Discriminant

    x² + 2x + 5 = 0. What is the discriminant?

    Answer