Roots of Square Equation Calculator

How to Use This Calculator

To find the roots of a quadratic equation:

1. Enter the coefficients a, b, and c in the input fields
2. Click the "Calculate" button
3. View the results showing the roots and discriminant
4. Interpret the results based on the discriminant value

The calculator will display the roots and provide information about the nature of the solutions based on the discriminant.

The Quadratic Formula

The roots of a quadratic equation ax² + bx + c = 0 can be found using the quadratic formula:

Where:

• a, b, and c are coefficients of the quadratic equation
• √(b² - 4ac) is the discriminant
• The ± symbol indicates there are two possible solutions

The discriminant determines the nature of the roots:

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

Understanding the Discriminant

The discriminant (D) is calculated as:

The discriminant provides important information about the roots:

• Positive discriminant: The equation has two real and distinct roots
• Zero discriminant: The equation has exactly one real root (a repeated root)
• Negative discriminant: The equation has no real roots, but two complex conjugate roots

Understanding the discriminant helps determine the nature of the solutions without actually calculating them.

Worked Examples

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0

Using the quadratic formula:

Solutions: x = 3 and x = 2

Example 2: One Real Root

Solve x² - 6x + 9 = 0

Using the quadratic formula:

Solution: x = 3 (repeated root)

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

Using the quadratic formula:

Solutions: x = -1 + 2i and x = -1 - 2i