Quadrtic Root 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.

What is a Quadratic Root?

A quadratic root is a solution to a quadratic equation. Quadratic equations are second-degree polynomials that can have zero, one, or two real roots. The roots represent the points where the parabola represented by the equation intersects the x-axis.

The standard form of a quadratic equation is:

Quadratic Equation

ax² + bx + c = 0

Where:

a, b, and c are coefficients
a ≠ 0 (otherwise it's not quadratic)

The roots can be found using the quadratic formula, which is derived from completing the square.

How to Use the Calculator

Using the quadratic root calculator is simple:

Enter the coefficients a, b, and c in the input fields
Click the "Calculate" button
View the results including the roots and discriminant
Use the chart to visualize the quadratic function

The calculator will show you the roots and provide additional information about the equation.

Quadratic Formula

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

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

Where:

x represents the roots
√(b² - 4ac) is the discriminant

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 roots

Worked Examples

Example 1: Two Distinct Real Roots

Solve x² - 5x + 6 = 0

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2

Roots: x = 3 and x = 2

Example 2: One Real Root

Solve x² - 6x + 9 = 0

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

Root: x = 3 (repeated)

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

Using the quadratic formula:

x = [-2 ± √(4 - 20)] / 2 = [-2 ± √(-16)] / 2

Roots: x = -1 ± 2i

Frequently Asked Questions

What is the difference between roots and coefficients?

Coefficients (a, b, c) are the numbers in the quadratic equation. Roots are the solutions to the equation that make it equal to zero.

Can quadratic equations have complex roots?

Yes, when the discriminant is negative, the roots will be complex numbers involving the imaginary unit i.

What does the discriminant tell us about the roots?

The discriminant (b² - 4ac) indicates the nature of the roots: positive for two real roots, zero for one real root, and negative for complex roots.