Quadratic Equation Solver

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, and a ≠ 0. The term "quadratic" comes from "quadratus," the Latin word for square, because the variable is squared.

The solutions to a quadratic equation are called roots, and they represent the x-values where the parabola (the graph of the quadratic function) crosses the x-axis. A quadratic equation can have two real roots, one repeated real root, or two complex conjugate roots.

How to Solve Quadratic Equations

There are several methods to solve quadratic equations:

Quadratic Formula: The most universal method, using x = (-b ± √(b² - 4ac)) / 2a
Factoring: Works when the quadratic can be factored into two binomials
Completing the Square: Rewriting the equation in the form (x + p)² = q
Graphing: Finding where the parabola intersects the x-axis

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

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

If b² - 4ac > 0: Two distinct real roots
If b² - 4ac = 0: One repeated real root
If b² - 4ac < 0: Two complex conjugate roots

Example

Standard Quadratic Example

Problem: Solve x² - 5x + 6 = 0

Solution:

Identify: a = 1, b = -5, c = 6
Discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
x = (5 ± √1) / 2 = (5 ± 1) / 2
x₁ = (5 + 1) / 2 = 3
x₂ = (5 - 1) / 2 = 2
Solutions: x = 3 or x = 2

Frequently Asked Questions

What is the discriminant?

The discriminant is the expression b² - 4ac under the square root in the quadratic formula. It tells us how many and what type of roots the equation has.

What if a = 0?

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. Use our Solve for X calculator for linear equations.

Can quadratic equations have complex roots?

Yes! When the discriminant is negative, the roots are complex numbers involving the imaginary unit i. For example, x² + 1 = 0 has roots x = ±i.

Solve ax² + bx + c = 0

Result