Algebra Calculator

Quadratic Equation Calculator

Solve equations in the form ax² + bx + c = 0. Find real or complex roots, the discriminant, vertex, axis of symmetry, and a clear step-by-step solution.

Solve a Quadratic Equation

Enter the coefficients a, b, and c.

Real & complex roots

ax² + bx + c = 0

Coefficient a

Must not be zero.

Coefficient b

Coefficient c

Your solution will appear here

Enter the three coefficients and click Solve Equation to calculate the roots and equation details.

Roots

Discriminant (Δ)

Axis of symmetry

Vertex

Y-intercept

Step-by-step solution

What This Quadratic Calculator Shows

Get more than just x-values. The calculator explains the most useful properties of the quadratic equation.

Real or Complex Roots

Automatically detects whether the equation has two real roots, one repeated root, or two complex roots.

Vertex and Axis

Calculates the parabola vertex and the vertical line that divides the graph symmetrically.

Step-by-Step Working

See the discriminant and quadratic formula substitution so you can follow how the answer is obtained.

How to Solve a Quadratic Equation

A quadratic equation is commonly written as ax² + bx + c = 0, where a, b, and c are constants and a cannot be zero.

x = (-b ± √(b² - 4ac)) / 2a Quadratic formula

What Is the Discriminant?

The expression b² - 4ac is called the discriminant. It tells you what type of roots the equation has.

Δ > 0: two distinct real roots.
Δ = 0: one repeated real root.
Δ < 0: two complex conjugate roots.

Example

For x² - 3x + 2 = 0, a = 1, b = -3, and c = 2. The discriminant is 1, so the equation has two real roots: x = 1 and x = 2.

When Is a Quadratic Equation Used?

Quadratic equations appear in algebra, geometry, physics, projectile motion, optimization, engineering, economics, and many other mathematical applications.

Quadratic Equation Calculator FAQs

Common questions about roots, discriminants, and quadratic equations.

What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation that can be written as ax² + bx + c = 0, where a is not zero.

Can a quadratic equation have no real roots?

Yes. If the discriminant is negative, the equation has two complex roots instead of real roots.

What happens when the discriminant equals zero?

The equation has one repeated real root, also called a double root.

Why can coefficient a not be zero?

If a equals zero, the x² term disappears and the equation becomes linear rather than quadratic.

Does this calculator support decimal coefficients?

Yes. You can enter integers, negative numbers, or decimal values for a, b, and c.