How this calculation works
A positive discriminant gives three real roots, while a negative discriminant gives one real root and a complex conjugate pair.
THE FORMULA
Δ = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcdA worked example
For x³−6x²+11x−6, Δ=4 and all three roots are distinct and real.
Good to know
The sign can be sensitive to rounding near repeated roots.
Reference: OpenStax Algebra and Trigonometry (opens in a new tab) ↗
Read the result
Classify roots of a real cubic using the sign of its discriminant.
The primary result is labeled “Discriminant”. Keep the input definitions in view when comparing this result with another method.
Try a related calculation
The sign can be sensitive to rounding near repeated roots.
For a related exercise, try the quadratic equation solver.
A few questions, answered
How can I check the default inputs by hand?
For x³−6x²+11x−6, Δ=4 and all three roots are distinct and real.
Which rule does this calculator apply?
Δ = b²c² − 4ac³ − 4b³d − 27a²d² + 18abcd