Certified calculator · Version 1

Polynomial Calculator

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

a (x³)b (x²)c (x)d

Distinct real root(s)1, 2, 3

Cubic discriminant-0.04

Trend

Included with Free

3.51.88

3.84.03

4.17.16

4.411.42

4.716.98

524

Visual scale

Accessible data view

1.7

2.3

2.6

2.9

3.2

3.5

3.8

4.1

4.4

4.7

CalcuMint Pro

Advanced Analysis

Compare plans

Advanced chartsExplore the result across time and assumptions.

Scenario comparisonCompare multiple assumptions side by side.

Sensitivity analysisSee which assumptions change the outcome most.

Goal SolverSolve backwards from the outcome you want.

Result methodology

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

Solution steps

Normalize cubic by a=1
Depressed cubic: p=-1, q=0
Discriminant Δ=-0.037037037037037035
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.

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.