Cubic Equation Solver (Cardano/Shengjin)
Cubic Equation Solver (Cardano/Shengjin): Applies Cardano's method to resolve cubic polynomial roots.
About this cubic equation solver (cardano/shengjin)
Cubic Equation Solver (Cardano/Shengjin) — browser-based utility.
How this tool works
Calculation and parameter relationship: a×x³+b×x²+c×x+d=0; (x−1)(x−2)(x−3)=x³−6x²+11x−6
- Paste the sample input into “Input”.
- Click “Run locally”.
- Compare “Raw Output Data” with the example; revise inputs when an error is shown.
Worked example
Scenario: Solve a cubic with three real integer roots after numeric precision normalization.
Sample input:
Processing: Reduce the cubic and evaluate its real-root branch; normalize only a real result within the ULP-scale integer bound.
Illustrative output:
Limits and verification
Parameter domains and units: (a,b,c,d) ∈ ℝ⁴; a ≠ 0; roots ∈ ℂ; binary64. Results use binary64 floating-point numbers unless BigInt is stated. Result cards show up to 10 significant digits; raw output retains the serialized value. A null ratio means the model has no meaningful finite value for that ratio; it is not zero.
Examples demonstrate an expected workflow; they do not prove every input or every branch of an external specification. Check important results with an independent source before using them for money, security, compliance, safety, or irreversible file changes.
Browser processing boundary
Tool input is processed by code running in the browser and is not intentionally sent to a CZOA processing API. The page can still request ordinary site assets, analytics, or advertising when those services are enabled. Browser extensions and managed-device software remain outside this tool's control.
Relevant references
These references govern or help explain the format, protocol, or calculation used here. Listing a reference does not claim certification or complete implementation of every optional feature.
- NIST Digital Library of Mathematical Functions, §1.2 — Elementary Algebra
- ECMAScript 2026, §6.1.6.1 Number Type
Content owner: CZOA Tools · Last reviewed: 2026-09-15 · Review methodology
How to use it
- Paste the sample input into “Input”.
- Click “Run locally”.
- Compare “Raw Output Data” with the example; revise inputs when an error is shown.
Frequently asked questions
How does the cubic solver compute its roots?+
It accepts four coefficients with nonzero leading coefficient, reduces the cubic to depressed-cubic terms, chooses its discriminant branch, and returns three re/im root objects. Real roots within a strict ULP-scale bound of an integer are normalized.
What input format does the page accept?+
Enter four whitespace-separated coefficients a b c d. The independently checked input is 1 -6 11 -6.
What precision and result boundaries apply?+
The solver uses JavaScript Number binary64 and trigonometric or cube-root branches; roots may be complex re/im objects. It is numeric evaluation, not a symbolic proof or exact algebraic-number system.
What result does the worked example produce?+
The corrected working source processed 1 -6 11 -6 and returned roots {re:3,im:0}, {re:1,im:0}, and {re:2,im:0}. the tool predates this precision correction.