Roots of A Quadratic 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 standard form ax² + bx + c = 0.

How to Use This Calculator

To find the roots of a quadratic equation, simply enter the coefficients a, b, and c from your equation into the calculator. The calculator will then compute the roots using the quadratic formula.

The calculator provides:

The two roots of the equation (if they exist)
The discriminant value
A visual representation of the quadratic function

You can also reset the calculator to start fresh or use the default values provided.

The Quadratic Formula

The standard form of a quadratic equation is:

ax² + bx + c = 0

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

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are coefficients
√(b² - 4ac) is the discriminant

The formula gives two solutions because a quadratic equation can have up to two roots.

Understanding 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 conjugate roots

The calculator will indicate which case applies based on the discriminant value.

Worked Examples

Example 1: 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 = 3 and (5 - 1)/2 = 2

Example 2: 2x² + 4x + 2 = 0

a = 2, b = 4, c = 2

Discriminant = 4² - 4(2)(2) = 16 - 16 = 0

Root: x = [-4 ± √0]/4 = -4/4 = -1 (double root)

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

How many roots can a quadratic equation have?

A quadratic equation can have two real roots, one real root (a repeated root), or two complex conjugate roots, depending on the discriminant.

What is the discriminant?

The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It determines the nature and number of roots of the quadratic equation.