Calculated Results
Results appear here after you solve the cubic.
| Value | Result |
|---|---|
| Root 1 | - |
| Root 2 | - |
| Root 3 | - |
| Depressed Cubic p | - |
| Depressed Cubic q | - |
| Sum of Roots | - |
| Product of Roots | - |
Calculator Inputs
Polynomial Graph
Example Data Table
| Equation | a | b | c | d | Typical Root Set |
|---|---|---|---|---|---|
| x³ - 6x² + 11x - 6 = 0 | 1 | -6 | 11 | -6 | 1, 2, 3 |
| x³ - 3x² + 3x - 1 = 0 | 1 | -3 | 3 | -1 | 1, 1, 1 |
| x³ + x + 1 = 0 | 1 | 0 | 1 | 1 | One real, two complex |
| 2x³ - 4x² - 22x + 24 = 0 | 2 | -4 | -22 | 24 | -3, 1, 4 |
Formula Used
General cubic: ax³ + bx² + cx + d = 0, where a ≠ 0.
Shift variable: Let x = t - b/(3a). This removes the square term and gives the depressed cubic:
t³ + pt + q = 0
p = (3ac - b²) / (3a²)
q = (27a²d - 9abc + 2b³) / (27a³)
Cardano discriminant:
Δ = (q/2)² + (p/3)³
- If Δ > 0, one real root and one complex pair appear.
- If Δ = 0, repeated roots appear.
- If Δ < 0, three distinct real roots appear.
How to Use This Calculator
- Enter the four coefficients for ax³ + bx² + cx + d = 0.
- Keep a nonzero, because zero changes the equation type.
- Set graph limits if you want a custom viewing window.
- Press Solve Cubic Equation to compute roots and polynomial checks.
- Review the displayed root pattern, discriminant, depressed cubic values, and verification row.
- Inspect the graph to see intercepts and turning behavior.
- Use the CSV or PDF buttons to save the displayed results.
Frequently Asked Questions
1. What does this calculator solve?
It solves cubic equations of the form ax³ + bx² + cx + d = 0. It reports real or complex roots, depressed cubic values, discriminant behavior, a verification check, and a graph of the polynomial.
2. Can cubic equations have complex roots?
Yes. A cubic always has three roots when multiplicity is counted. Some or all may be real, and a nonreal pair appears as complex conjugates when the coefficients are real.
3. What does the discriminant tell me?
The discriminant classifies the root pattern. Positive means one real root and one complex pair. Zero indicates repetition. Negative means three distinct real roots.
4. Why does the calculator convert to a depressed cubic?
The substitution x = t - b/(3a) removes the squared term. That simpler form makes Cardano-based solving easier and gives a cleaner path for analyzing discriminant cases.
5. What if coefficient a equals zero?
Then the equation is not cubic. It becomes quadratic, linear, or constant. This calculator flags that input because cubic root formulas require a nonzero leading coefficient.
6. Are repeated roots shown clearly?
Yes. When the discriminant is near zero, the results label the pattern as repeated or triple roots. Close numerical values are also rounded consistently for easier comparison.
7. Why might the graph window miss a root?
A root can fall outside the chosen x-range. Expand the graph limits or use the default example values to inspect a wider domain and reveal intercepts more clearly.
8. Can I save the results?
Yes. The page includes CSV export for spreadsheet use and PDF export for reports or study notes. Both downloads use the current displayed results.