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Formula verified · Deterministic calculation

Polynomial Calculator

Solve ax³+bx²+cx+d=0 with a deterministic cubic engine using depressed-cubic and Cardano/trigonometric identities.

Distinct real root(s)1, 1.9999999999999998, 3
Cubic discriminant
-0.04
Methodology

The certified cubic engine normalizes the polynomial and uses deterministic Cardano or trigonometric identities according to its discriminant.

Sources
  • CalcuMint deterministic polynomial engine
Solution steps
  1. Normalize cubic by a=1
  2. Depressed cubic: p=-1, q=0
  3. Discriminant Δ=-0.037037037037037035
  4. Real root(s): 1, 1.9999999999999998, 3

How this calculator works

The cubic is normalized and transformed to t³+pt+q=0. Its discriminant determines whether the real-root branch uses Cardano radicals or the trigonometric identity.

Formula

ax³+bx²+cx+d=0 → depressed cubic t³+pt+q=0

Finds the distinct real roots of a cubic polynomial using deterministic Cardano/trigonometric identities.

Worked example

x³ - 1 is a verified test case used by the calculation engine.

Assumptions and limitations

  • The leading cubic coefficient must be non-zero.
  • This certified version reports distinct real roots; non-real cubic roots are not displayed.

Methodology & sources

This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint. Technical calculation version: 1.

  • CalcuMint deterministic polynomial engine — Standard cubic reduction and Cardano identities.

Frequently asked questions

What degree is supported?

This version solves cubic polynomials.

Does it return complex cubic roots?

Not in this certified version; it reports the distinct real roots.

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