Cubic Polynomial Roots Calculator

Calculator Input

Coefficient a

Must not be zero.

Coefficient b

Coefficient c

Coefficient d

Decimal precision

Graph sample points

Graph x minimum

Graph x maximum

Show method details

Example Data Table

Equation	Coefficients (a, b, c, d)	Root Pattern	Example Roots
x³ - 6x² + 11x - 6 = 0	(1, -6, 11, -6)	Three distinct real roots	1, 2, 3
x³ - 3x + 2 = 0	(1, 0, -3, 2)	Repeated real roots	1, 1, -2
x³ + x + 1 = 0	(1, 0, 1, 1)	One real and two complex roots	Approximately -0.6823, 0.3412 ± 1.1615i

Formula Used

General cubic: ax³ + bx² + cx + d = 0

Normalize: divide by a, where a ≠ 0.

Shift variable: x = t - b / (3a)

Depressed cubic: t³ + pt + q = 0

p: (3ac - b²) / (3a²)

q: (27a²d - 9abc + 2b³) / (27a³)

Discriminant test: Δ = (q/2)² + (p/3)³

When Δ is positive, the equation has one real root and one complex conjugate pair.

When Δ is zero, the equation has repeated real roots.

When Δ is negative, the equation has three distinct real roots.

How to Use This Calculator

Enter the four coefficients for the cubic equation.
Set the decimal precision for displayed values.
Adjust graph range and sample points if needed.
Enable method details to see intermediate values.
Click Calculate Roots to generate the results.
Review the roots, discriminant, and graph below the header.
Download the result table as CSV.
Use the PDF button to save a printable report.

Frequently Asked Questions

1. What does this calculator solve?

It solves cubic equations of the form ax³ + bx² + cx + d = 0. It returns real or complex roots, root type, discriminant values, and a visual graph.

2. What happens if a equals zero?

If a is zero, the equation is not cubic. This calculator requires a nonzero leading coefficient, so it will show a validation error instead.

3. Can it show complex roots?

Yes. When the cubic has nonreal solutions, the calculator displays the complex conjugate pair using standard a + bi notation.

4. Why do I see repeated roots?

Repeated roots appear when the discriminant condition indicates multiple roots at the same x-value. This often happens when the curve touches the x-axis and turns.

5. Does the graph include all roots?

The graph shows the polynomial curve on the chosen x-range. Real roots appear where the curve crosses or touches the x-axis. Complex roots are listed in the results table.

6. What is the role of p, q, and Δ?

They come from converting the original equation into a depressed cubic. These values help classify the roots and determine which Cardano-based solution path to use.

7. Can I export my results?

Yes. You can download a CSV file containing the computed roots and key values. You can also save a PDF report from the generated result section.

8. How accurate are the answers?

The calculator uses floating-point math and lets you choose display precision. For most practical uses, it is highly accurate, though very sensitive inputs may show rounding effects.