Real Imaginary Roots Calculator

Introduction

Quadratic equations are fundamental in mathematics and appear in various scientific and engineering applications. The roots of a quadratic equation can be either real or complex, depending on the discriminant. This calculator determines both types of roots for any quadratic equation of the form:

The roots are calculated using the quadratic formula, which is derived from completing the square. The discriminant (Δ) determines the nature of the roots:

• If Δ > 0: Two distinct real roots
• If Δ = 0: One real root (a repeated root)
• If Δ < 0: Two complex conjugate roots

How to Use the Calculator

1. Enter the coefficients a, b, and c of your quadratic equation in the input fields.
2. Click the "Calculate" button to compute the roots.
3. View the results in the output section below the calculator.
4. Use the "Reset" button to clear all inputs and results.

Formula

The roots of a quadratic equation ax² + bx + c = 0 are given by:

Where:

• a, b, c are the coefficients of the quadratic equation
• √(b² - 4ac) is the square root of the discriminant

For complex roots, the square root of a negative number is expressed using the imaginary unit i, where i² = -1.

Example Calculation

Let's find the roots of the equation x² - 5x + 6 = 0:

1. Identify the coefficients: a = 1, b = -5, c = 6
2. Calculate the discriminant: Δ = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
3. Since Δ > 0, there are two distinct real roots
4. Apply the quadratic formula:

   x = [5 ± √1] / 2 = [5 ± 1] / 2

5. Calculate the roots:
   • x₁ = (5 + 1)/2 = 3
   • x₂ = (5 - 1)/2 = 2

The roots of the equation x² - 5x + 6 = 0 are 3 and 2.

Interpreting Results

The calculator provides three possible outcomes based on the discriminant:

Case 1: Two Real Roots (Δ > 0)

This indicates the quadratic equation crosses the x-axis at two distinct points. The roots are real and distinct.

Case 2: One Real Root (Δ = 0)

This means the quadratic equation touches the x-axis at exactly one point. The root is real and repeated.

Case 3: Complex Roots (Δ < 0)

When the discriminant is negative, the equation does not intersect the x-axis. The roots are complex conjugates, expressed as a + bi and a - bi, where i is the imaginary unit.