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Quadratic Equation Complex Roots Calculator

Reviewed by Calculator Editorial Team

This calculator finds the complex roots of quadratic equations of the form ax² + bx + c = 0. When the discriminant is negative, the roots are complex numbers. The calculator provides both the roots and visualizes them on the complex plane.

Introduction

A quadratic equation is a second-degree polynomial equation of the form:

Quadratic Equation

ax² + bx + c = 0

Where a, b, and c are coefficients, and x represents the variable. The roots of the equation are the values of x that satisfy the equation.

When the discriminant (b² - 4ac) is negative, the equation has two complex roots. Complex roots are solutions that involve the imaginary unit i, where i² = -1.

Formula

The roots of a quadratic equation can be found using the quadratic formula:

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

When the discriminant (D = b² - 4ac) is negative, the roots are complex and can be expressed as:

Complex Roots

x₁ = [-b + √(4ac - b²)i] / (2a)

x₂ = [-b - √(4ac - b²)i] / (2a)

Where √(4ac - b²) represents the square root of the positive discriminant (4ac - b²).

How to Use the Calculator

  1. Enter the coefficients a, b, and c in the calculator form.
  2. Click the "Calculate" button to find the roots.
  3. View the results in the result panel below the calculator.
  4. Use the chart to visualize the roots on the complex plane.
  5. Click "Reset" to clear the form and start over.

Note

The calculator assumes a ≠ 0. If a = 0, the equation is no longer quadratic.

Worked Example

Let's find the complex roots of the equation x² + 2x + 5 = 0.

  1. Identify the coefficients: a = 1, b = 2, c = 5.
  2. Calculate the discriminant: D = b² - 4ac = 4 - 20 = -16.
  3. Since D is negative, the roots are complex.
  4. Apply the quadratic formula:

    5. Calculation

    x = [-2 ± √(-16)] / 2

    x = [-2 ± 4i] / 2

    x₁ = -1 + 2i

    x₂ = -1 - 2i

  5. The complex roots are -1 + 2i and -1 - 2i.

Interpreting Results

The calculator provides the roots in the form of complex numbers. Each root consists of a real part and an imaginary part:

  • Real part: The coefficient of the real number.
  • Imaginary part: The coefficient of i, where i is the imaginary unit.

The chart visualizes the roots on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part.

FAQ

What is a complex root?
A complex root is a solution to a quadratic equation that involves the imaginary unit i, where i² = -1. Complex roots occur when the discriminant is negative.
How do I know if a quadratic equation has complex roots?
A quadratic equation has complex roots if the discriminant (b² - 4ac) is negative. The calculator checks this condition and provides complex solutions accordingly.
Can the calculator handle non-integer coefficients?
Yes, the calculator accepts any real number as coefficients a, b, and c. It will compute the roots accurately for both integer and non-integer values.
What does the chart show?
The chart displays the complex roots on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part of the roots.
Is the quadratic formula always accurate?
Yes, the quadratic formula is a mathematically proven method for finding the roots of any quadratic equation, whether the roots are real or complex.