Solve ax3 + bx2 + cx + d = 0
Enter coefficients below to find all roots (real and complex) of your cubic polynomial.
How to use this cubic equations calculator
This tool solves polynomial equations of the form ax3 + bx2 + cx + d = 0. Just enter the four coefficients, click Calculate Roots, and the calculator returns all solutions.
- If the equation has three real roots, all three are shown.
- If it has one real root and two complex roots, the complex pair is displayed in a ± bi form.
- The discriminant is provided so you can quickly identify the root pattern.
What is a cubic equation?
A cubic equation is a third-degree polynomial equation. The highest exponent on x is 3. Cubics are important because they are the next step beyond quadratic equations and appear in many applied math and engineering problems.
General form
ax3 + bx2 + cx + d = 0, where a ≠ 0.
When a = 0, the equation is no longer cubic. This calculator gracefully falls back to quadratic or linear solving so you still get an answer.
How the solver works (plain-English overview)
The calculator uses a standard algebraic approach based on the depressed cubic transformation and Cardano-style formulas. In short:
- It transforms the original cubic to remove the squared term.
- It computes reduced parameters usually called p and q.
- It evaluates the discriminant to determine whether roots are real, repeated, or complex.
- It calculates all roots and formats them for easy reading.
This method is exact in symbolic math and highly accurate numerically for normal coefficient ranges.
Understanding the discriminant
The cubic discriminant shown in the result tells you what kind of roots to expect:
- Δ > 0: one real root and two non-real complex conjugate roots.
- Δ = 0: all roots are real and at least two are equal (repeated root).
- Δ < 0: three distinct real roots.
Worked examples
Example 1: x3 - 1 = 0
Set a=1, b=0, c=0, d=-1. You get one real root at x=1 and two complex roots: -0.5 ± 0.8660i.
Example 2: x3 - 6x2 + 11x - 6 = 0
Set a=1, b=-6, c=11, d=-6. This factors as (x-1)(x-2)(x-3), so roots are 1, 2, and 3.
Example 3: 2x3 + 4x2 - 6x = 0
Set a=2, b=4, c=-6, d=0. One root is obvious: x=0. The remaining quadratic factor gives the other real roots.
Where cubic equations appear in real life
- Physics (motion equations and energy relationships)
- Engineering design and control systems
- Computer graphics and geometric intersections
- Economics and optimization models
- Data fitting and polynomial regression
Common mistakes to avoid
- Sign errors: entering + instead of - for coefficients is the #1 issue.
- Forgetting that a must be nonzero: if a=0, it is not a cubic.
- Rounding too early: keep full precision while interpreting results.
- Ignoring complex roots: many cubics naturally produce them.
FAQ
Does this calculator show complex roots?
Yes. Complex roots are shown in standard real ± imaginary i format.
Can I enter decimals or negative numbers?
Absolutely. The input fields accept decimals, negative values, and large magnitudes.
What if my equation is quadratic or linear?
The script detects that case automatically and solves the reduced equation type.
Final note
This cubic equations calculator is useful for homework checks, quick research calculations, and practical engineering math. If you need symbolic factorization, graphing, or step-by-step derivations, pair this with a CAS tool—but for fast numeric roots, this page should cover most needs.