Roots Quadratic Equation Online Calculator

Quadratic equations are fundamental in algebra and appear in many real-world problems. This calculator helps you find the roots of any quadratic equation in the form ax² + bx + c = 0, whether they're real or complex.

What is a Quadratic Equation?

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

ax² + bx + c = 0

Where:

a, b, and c are constants
a ≠ 0 (if a = 0, it's a linear equation)
x is the variable we're solving for

The solutions to this equation are called roots or zeros. A quadratic equation can have:

Two distinct real roots
One real root (a repeated root)
Two complex conjugate roots

How to Use This Calculator

Enter the coefficients a, b, and c in the input fields
Click "Calculate Roots" to find the solutions
View the results and graph visualization
Use the "Reset" button to clear all inputs

Note: For complex roots, the calculator will display them in the form x = a ± bi where i is the imaginary unit (√-1).

The Quadratic Formula

The standard method for solving quadratic equations is the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

The discriminant (b² - 4ac) 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

Worked Examples

Example 1: Two Real Roots

Solve x² - 5x + 6 = 0

a = 1, b = -5, c = 6

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1

Roots: x = [5 ± √1]/2 = (5 ± 1)/2

x₁ = (5 + 1)/2 = 3

x₂ = (5 - 1)/2 = 2

Example 2: One Real Root

Solve x² - 6x + 9 = 0

a = 1, b = -6, c = 9

Discriminant = (-6)² - 4(1)(9) = 36 - 36 = 0

Root: x = [6 ± √0]/2 = 6/2 = 3

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

a = 1, b = 2, c = 5

Discriminant = 2² - 4(1)(5) = 4 - 20 = -16

Roots: x = [-2 ± √-16]/2 = [-2 ± 4i]/2

x₁ = -1 + 2i

x₂ = -1 - 2i

Frequently Asked Questions

What is the difference between roots and coefficients?: The coefficients (a, b, c) are the numbers in the quadratic equation. The roots are the solutions to the equation.
Can quadratic equations have more than two roots?: No, quadratic equations can have at most two roots (real or complex).
What does a negative discriminant mean?: A negative discriminant indicates the equation has two complex conjugate roots.
How do I know if my equation is quadratic?: An equation is quadratic if the highest power of the variable is 2 (like x²).
Can this calculator solve higher-degree equations?: No, this calculator is specifically designed for quadratic equations (degree 2).