What does the discriminant tell me?

The discriminant is b² - 4ac. Above zero gives two real roots, zero gives one repeated real root, and below zero gives a conjugate pair of complex roots. It is shown beside the roots so you can check which case was used.

What happens when a is zero?

The equation is no longer quadratic. When b is non-zero, the calculator solves the resulting linear equation. When b is also zero, it reports either no solution or every real number explicitly instead of applying the quadratic formula.

How does the vertex-form page work?

Open the vertex-form page to convert ax² + bx + c to a(x-h)² + k, where h = -b/(2a) and k = c - b²/(4a). It shows the vertex and a small parabola diagram. A zero a coefficient is rejected because it is not a parabola.

The inputs are finite decimal numbers and the displayed roots use JavaScript number arithmetic, so they are numerical approximations. Every root includes a normalized substitution residual; a value close to zero indicates that the displayed root satisfies the entered equation within numerical precision.

Does my equation leave this browser?

No. The calculation uses the values in this tab. Only the coefficient settings are optionally kept in this browser's local storage, and blocked storage does not stop the calculator. Coefficients are not sent to analytics.

Quadratic Formula Calculator

Enter a, b and c to solve ax² + bx + c = 0. The calculator shows the discriminant, handles linear and constant cases explicitly, checks each root by substitution, and can convert the same quadratic to vertex form.

Result1, 2two real

Common questions

What does the discriminant tell me?: The discriminant is b² - 4ac. Above zero gives two real roots, zero gives one repeated real root, and below zero gives a conjugate pair of complex roots. It is shown beside the roots so you can check which case was used.
What happens when a is zero?: The equation is no longer quadratic. When b is non-zero, the calculator solves the resulting linear equation. When b is also zero, it reports either no solution or every real number explicitly instead of applying the quadratic formula.
How does the vertex-form page work?: Open the vertex-form page to convert ax² + bx + c to a(x-h)² + k, where h = -b/(2a) and k = c - b²/(4a). It shows the vertex and a small parabola diagram. A zero a coefficient is rejected because it is not a parabola.
Are the roots exact?: The inputs are finite decimal numbers and the displayed roots use JavaScript number arithmetic, so they are numerical approximations. Every root includes a normalized substitution residual; a value close to zero indicates that the displayed root satisfies the entered equation within numerical precision.
Does my equation leave this browser?: No. The calculation uses the values in this tab. Only the coefficient settings are optionally kept in this browser's local storage, and blocked storage does not stop the calculator. Coefficients are not sent to analytics.

Uses the stable quadratic formulation for finite decimal coefficients up to 10^100. Roots and the discriminant are approximately represented in JavaScript, and each result includes a normalized substitution check; this calculator describes algebra and gives no advice about what a result means.