Calculator Inputs
Formula Used
How to Use This Calculator
- Enter the four coefficients for your cubic equation.
- Keep coefficient a nonzero to preserve cubic behavior.
- Choose the decimal precision for displayed results.
- Optionally enter custom graph limits for the x-axis.
- Press Solve Cubic Equation to calculate roots and diagnostics.
- Review the roots, discriminant, inflection point, and verification values.
- Inspect the Plotly graph to see crossings and turning points.
- Use the CSV or PDF buttons to export the solution summary.
Example Data Table
| Example | a | b | c | d | Expected root behavior |
|---|---|---|---|---|---|
| Classic factorable cubic | 1 | -6 | 11 | -6 | Three real roots: 1, 2, and 3 |
| One real and two complex | 1 | 0 | 1 | 1 | One real root with a complex conjugate pair |
| Repeated real root case | 1 | -3 | 3 | -1 | Triple real root at x = 1 |
FAQs
1. What does this calculator solve?
It solves cubic equations of the form ax³ + bx² + cx + d = 0. It returns real roots, complex roots, discriminants, and a graph.
2. Why must coefficient a be nonzero?
If a equals zero, the equation is no longer cubic. It becomes quadratic or lower, which requires a different solving method.
3. What does the discriminant tell me?
The discriminant shows the root pattern. Positive means one real root, zero means repeated real roots, and negative means three distinct real roots.
4. Can the calculator show complex roots?
Yes. When the cubic has nonreal solutions, the output lists the complex conjugate pair using standard a ± bi notation.
5. What is the Plotly graph useful for?
The graph helps you see x-axis crossings, turning points, and the overall curve shape. It is useful for checking whether numerical results match visual behavior.
6. Are repeated roots handled correctly?
Yes. The solver checks the special case where the discriminant is zero and reports equal roots or a double root plus a simple root.
7. What do p and q represent?
They are the depressed cubic parameters after removing the x² term. Cardano’s method uses them directly to classify and compute roots.
8. Can I export the results?
Yes. You can download a CSV summary for spreadsheet use or create a PDF report for sharing, printing, or record keeping.