Then the equation isn't actually quadratic — it's linear (bx + c = 0), and should be solved differently.

What are complex roots used for?

Complex roots come up throughout engineering and physics (like signal processing and control systems), even though they don't correspond to points where the parabola crosses the x-axis.

Quadratic Formula Calculator

Solve any quadratic equation of the form ax² + bx + c = 0 using the quadratic formula, including equations with complex roots.

Two Real Roots

x = 3, x = 2

Discriminant: 1

How It Works

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a. The expression under the square root, b² − 4ac, is called the discriminant.

What the discriminant tells you

A positive discriminant means two distinct real roots, zero means one repeated real root, and a negative discriminant means the two roots are complex numbers (involving i, the imaginary unit).

Example

For x² − 5x + 6 = 0, the discriminant is 1, giving two real roots: x = 3 and x = 2.

Common Mistakes

Forgetting the ± in the formula — every quadratic equation with a positive discriminant has two solutions, not one.
Assuming a negative discriminant means 'no solution' — it means no real solution, but there are still two valid complex solutions.

Frequently Asked Questions

What if a is 0?: Then the equation isn't actually quadratic — it's linear (bx + c = 0), and should be solved differently.
What are complex roots used for?: Complex roots come up throughout engineering and physics (like signal processing and control systems), even though they don't correspond to points where the parabola crosses the x-axis.