Root Quadratic Equation Calculator

Find the roots of any quadratic equation with our root quadratic equation calculator. This tool uses the quadratic formula to solve for x in equations of the form ax² + bx + c = 0. Learn how to interpret the results, understand the discriminant, and visualize solutions with our interactive chart.

How to Use This Calculator

Using our root quadratic equation calculator is simple:

Enter the coefficients a, b, and c from your quadratic equation in the input fields.
Click the "Calculate" button to solve for the roots.
View the results, including the roots and discriminant.
Use the interactive chart to visualize the quadratic function.

The calculator will display the roots in both decimal and exact form when possible. For complex roots, it will show the real and imaginary parts separately.

The Quadratic Formula

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

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0
√(b² - 4ac) is the discriminant
The ± symbol indicates there are two possible solutions

The quadratic formula works for all quadratic equations where a ≠ 0. If a = 0, the equation is no longer quadratic and should be solved using linear methods.

Understanding the Discriminant

The discriminant (b² - 4ac) determines the nature of the roots:

If the discriminant is positive, there are two distinct real roots.
If the discriminant is zero, there is exactly one real root (a repeated root).
If the discriminant is negative, there are two complex conjugate roots.

The calculator will indicate which case applies based on the values you enter. Understanding the discriminant helps you predict the type of solutions before calculating them.

Worked Examples

Example 1: Two Real Roots

Solve x² - 5x + 6 = 0

Using the quadratic formula:

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

Solutions: 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 ± 0] / 2

Solution: x = 3 (double root)

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

Using the quadratic formula:

x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2

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

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.
When should I use the quadratic formula?: Use the quadratic formula when you need to solve a quadratic equation and completing the square or factoring would be too difficult or time-consuming.
What does the discriminant tell me?: The discriminant (b² - 4ac) tells you the nature of the roots: positive for two real roots, zero for one real root, and negative for two complex roots.
Can I solve quadratic equations with fractions?: Yes, you can enter fractional coefficients directly into the calculator. The quadratic formula will work with any real numbers for a, b, and c (as long as a ≠ 0).
What if I get an error when calculating?: If you see an error, check that you've entered valid numbers and that a is not zero. The calculator will show an error message explaining the issue.