Cubic Equation Solver (Cardano/Shengjin)
Solve cubic equations with exact discriminant classification, repeated-root multiplicities, three numerical roots and trigonometric or Cardano expressions.
About this cubic equation solver (cardano/shengjin)
Solve cubic equations with exact discriminant classification, repeated-root multiplicities, three numerical roots and trigonometric or Cardano expressions Computes exact algebraic and numerical metrics directly in your browser using IEEE 754 precision without round-trip latency.
Formula & calculation mechanics
The execution engine processes input tokens and applies transformations in the browser sandbox. Specifically, For ax³+bx²+cx+d=0, a≠0. A=b²−3ac, B=bc−9ad, C=c²−3bd, S=B²−4AC, and D=−S/3. D>0 (S<0): three distinct real roots. D<0 (S>0): one real root and a conjugate pair. D=0: a triple root if A=0; otherwise one simple and one double root.
Worked numerical calculation
Scenario: What does the example produce?
Sample input:
Processing: Enter four rational coefficients with a nonzero. Read the exact discriminants, root classification and multiplicities. Compare the three decimal roots with the exact algebraic or trigonometric expressions.
Illustrative output:
Domain range & precision boundaries
Each field accepts exact rational input: at most 64 UTF-16 code units, 30 mantissa digits including leading zeros or 30 digits per fraction integer; decimal exponent −30…30. a ≠ 0. These are browser workload limits, not usage quotas. Roots are repeated in the list according to their multiplicities. Discriminants, multiplicities and formula parameters use exact rational BigInt arithmetic. Numerical real roots are isolated with exact polynomial signs and bisection; a conjugate pair is recovered by polynomial division. There is no floating tolerance for deciding multiplicity. No added calculation dependency or external service. Decimals are rounded binary64 approximations, not exact numerical values. Nearby distinct roots may display the same digits; classification and symbolic expressions remain exact. No claim of arbitrary-precision decimal output. Angles are in radians; arccos takes its principal value in [0, π]. This real trigonometric form applies to three distinct real roots and avoids square roots of negative real numbers. The listed numerical roots are in ascending order. ∛ denotes the real cube root; both radicands are real, and uv=−p/3. i²=−1. The ± expression gives the conjugate pair as an unordered pair; numerical output lists the positive imaginary part first. The repeated roots are exact reduced fractions. For A=B=0, x=−b/(3a) has multiplicity three; otherwise the double root is −B/(2A), and the simple root is −b/a plus B/A. Runs locally in your browser, with no added calculation dependency, fee or external service. Available offline after loading.
Calculations in Cubic Equation Solver (Cardano/Shengjin) use standard IEEE 754 precision or exact rational models. Verify significant accounting or tax figures with authoritative references.
In-browser calculation boundary
Formula calculations and numeric evaluations in Cubic Equation Solver (Cardano/Shengjin) run synchronously in the browser JavaScript engine with zero server telemetry.
Mathematical formulas & precision standards
Mathematical constants, IEEE 754 floating-point standards, and numerical analysis specifications:
- NIST Digital Library of Mathematical Functions, §1.2 — Elementary Algebra
- ECMAScript 2026, §6.1.6.1 Number Type
Cubic Equation Solver (Cardano/Shengjin) verification · Audit standards
How to use it
- Enter, paste, or select your input data into the Cubic Equation Solver (Cardano/Shengjin) workspace controls.
- Review parameter fields, units, or options configured for Cubic Equation Solver (Cardano/Shengjin).
- Click the action button or observe immediate live calculations rendered by Cubic Equation Solver (Cardano/Shengjin).
Frequently asked questions
How are the roots classified?+
For ax³+bx²+cx+d=0, a≠0.
What do the formulas and decimal results mean?+
Boundary constraints concerning "What do the formulas and decimal results mean?" prevent invalid parameter state mutations client-side.
What are the input limits?+
Runtime handling of "What are the input limits?" executes synchronously in your browser tab without remote delays.
What does the example produce?+
Security isolation for "What does the example produce?" ensures zero transmission beyond your local workstation.