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Cubic Equation Solver — result sheet
Solve a real cubic equation for all three roots, including repeated and complex-conjugate roots, and classify the root pattern with the cubic discriminant.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: For ax³+bx²+cx+d=0, depress the cubic with x=u−b/(3a), solve u³+pu+q=0 using Cardano or the three-real-root trigonometric branch, and use Δ=b²c²−4ac³−4b³d−27a²d²+18abcd to classify roots.
This solver returns all three complex roots counted with multiplicity rather than only the real crossing points. It uses a real trigonometric branch when three real roots are present, a stable real cube-root branch when one root is real, and a repeated-root branch when the discriminant is numerically zero.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Coefficient a of x³ — minimum -1000000; maximum 1000000
- Coefficient b of x² — minimum -1000000; maximum 1000000
- Coefficient c of x — minimum -1000000; maximum 1000000
- Constant d — minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Coefficient a of x³ | 1 |
| Coefficient b of x² | -6 |
| Coefficient c of x | 11 |
| Constant d | -6 |
x³−6x²+11x−6 = 0 has three distinct real roots: 1, 2, and 3; its discriminant is 4.
Assumptions and limits
- The leading coefficient a is nonzero, so the entered equation is genuinely cubic.
- Coefficients are real finite numbers; displayed roots are rounded text approximations when they are not exact integers.
- The discriminant classifies the polynomial roots but does not replace a graph, factorization proof, or domain-specific interpretation.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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