MathsLive

Quadratic Equation Solver

Enter the coefficients a, b and c to solve ax² + bx + c = 0.

Short answer

Roots from the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac tells you if roots are real and distinct (positive), real and equal (zero) or complex (negative).

Coefficients

a (coefficient of x²)

b (coefficient of x)

c (constant)

Fill in the fields above to see your result instantly.

Calculation method, coverage and version

Inputs used: a (coefficient of x²), b (coefficient of x), c (constant).
Calculation approach: The calculator applies the rules and assumptions described in “How it works” to the values you enter. Rounding can cause small differences from an official or provider calculation.
Coverage: Maths. UK nations, tax treatment, local rules and provider criteria can differ; check the official sources shown below before acting.
Content version: No page-specific review date is currently published. Treat the result as indicative and verify current rules independently.

Not included unless explicitly requested: personal circumstances not entered, provider discretion, future rule changes, professional fees and case-specific exceptions. Read the full calculator methodology.

How it works

The quadratic formula x = (−b ± √(b² − 4ac)) / 2a always works. The discriminant b² − 4ac decides root type, and the vertex sits at x = −b/2a.

Worked example

x² − 3x + 2 = 0 → discriminant 1 → x = 2 or x = 1.

Who should use this

•GCSE / A-level maths students
•Engineers solving projectile equations

Common mistakes

×Forgetting the ± in the formula
×Sign errors when b is negative

Frequently asked questions

What does a complex root mean?▾

The parabola never crosses the x-axis. The roots involve √-1 (i).

What if a = 0?▾

It's no longer quadratic. Solve bx + c = 0 → x = -c/b.

Related calculators

Average / Mean →Standard Deviation →