Skip to content
Mom's Basement University

Complex Numbers in Quadratic Equations

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn how to solve quadratic equations that have a negative discriminant and express the solutions as complex numbers.

The Concept

Section titled “The Concept”

When the discriminant b² − 4ac is negative, a quadratic equation has no real solutions, but it does have complex solutions.

The quadratic formula still works:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

When the discriminant is negative, we write the square root using the imaginary unit i:

−k=k⋅i\sqrt{-k} = \sqrt{k} \cdot i

The solutions will be in the form a + bi and a − bi, always a conjugate pair (same real part, opposite imaginary parts).

Worked Example

Section titled “Worked Example”

Solve x² + 4x + 13 = 0.

The left panel shows the parabola y = x² + 4x + 13. It floats entirely above the x-axis. There are no x-intercepts, which means no real solutions. The right panel shows where the solutions actually live: on the complex plane, as the conjugate pair −2 + 3i and −2 − 3i. The dashed purple line connecting them shows they’re mirror images across the real axis.

1. Find the discriminant

Here a = 1, b = 4, c = 13.

D=42−4(1)(13)=16−52=−36D = 4^2 - 4(1)(13) = 16 - 52 = -36

D = −36 < 0, so the solutions are complex.

2. Apply the quadratic formula

x=−4±−362(1)=−4±36⋅i2=−4±6i2x = \frac{-4 \pm \sqrt{-36}}{2(1)} = \frac{-4 \pm \sqrt{36} \cdot i}{2} = \frac{-4 \pm 6i}{2}

3. Simplify

x=−42±6i2=−2±3ix = \frac{-4}{2} \pm \frac{6i}{2} = -2 \pm 3i

The solutions are −2 + 3i and −2 − 3i, a conjugate pair.

Notice that both solutions have the same real part (−2) and opposite imaginary parts (+3i and −3i). This always happens when the quadratic has real coefficients and a negative discriminant.

Real-World Application

Section titled “Real-World Application”

Complex solutions appear in many technical fields even when the final answer is interpreted in real terms:

  • Electrical engineering: analyzing alternating current (AC) circuits where voltage and current are out of phase.
  • Mechanical vibrations and resonance (damped harmonic motion).
  • Control systems and stability analysis.
  • Signal processing and audio engineering.

For example, when modeling the motion of a damped spring, complex numbers help describe how the system oscillates and loses energy over time.

Quiz

Section titled “Quiz”
Solve $x^2 + 6x + 13 = 0$
If the discriminant is negative, the solutions are:
What is $\sqrt{-49}$ in terms of $i$?
The solutions to a quadratic with negative discriminant always come in:
Solve $x^2 + 4x + 8 = 0$.