Cubic Equation Calculator
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A cubic equation ax³ + bx² + cx + d = 0 always has three roots, counted with multiplicity, and at least one is real. For x³ − 6x² + 11x − 6 = 0 the roots are 1, 2 and 3, because the cubic factors as (x − 1)(x − 2)(x − 3).
Real roots are where the curve crosses the x-axis.
| Roots | |
| x1 | 1 |
| x2 | 3 |
| x3 | 2 |
| Checks (Vieta) | |
| Sum of roots = −b/a | 6 |
| Product of roots = −d/a | 6 |
| Turning points at x | 2.57735 and 1.42265 |
Show steps
- Rational root theorem: any rational root is ±p/q with p dividing 6 and q dividing 1.
- Test the candidates: x = 1 makes the left side 0, so (x − 1) is a factor.
- Divide (synthetic division) to get the quadratic factor 1x² − 5x + 6.
- Solve 1x² − 5x + 6 = 0 with the quadratic formula: discriminant = -5² − 4(1)(6) = 1.
How to use the cubic equation calculator
- Write your equation as ax³ + bx² + cx + d = 0 (move everything to one side).
- Enter a, b, c and d. Use 0 for a missing term, and decimals are fine.
- Press Solve. You get the roots, the method, a check with Vieta’s formulas and a graph.
Method 1: rational root theorem and factoring
If the coefficients are integers, any rational root is ±p/q, where p divides d and q divides a. Test the candidates; when one works, divide by (x − root) to get a quadratic, then use the quadratic formula. Example: for 2x³ − 3x² − 11x + 6, x = 3 works, and 2x³ − 3x² − 11x + 6 = (x − 3)(2x² + 3x − 2) = (x − 3)(2x − 1)(x + 2).
Method 2: Cardano and the trigonometric method
Substitute x = t − b/(3a) to remove the x² term, giving t³ + pt + q = 0. The discriminant Δ = −(4p³ + 27q²) decides the case:
- Δ > 0: three different real roots (use the cosine formula).
- Δ = 0: a repeated root.
- Δ < 0: one real root and two complex conjugates (use Cardano’s formula).
Checking your roots
For roots r₁, r₂, r₃ of ax³ + bx² + cx + d: r₁ + r₂ + r₃ = −b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a and r₁r₂r₃ = −d/a. For x³ − 6x² + 11x − 6: 1 + 2 + 3 = 6 and 1 · 2 · 3 = 6.
Cubic equations practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you solve a cubic equation step by step?
Look for a rational root with the rational root theorem, divide it out with synthetic division, and solve the remaining quadratic. If there is no rational root, use Cardano's formula or the trigonometric method.
Is there a cubic formula?
Yes. Cardano's formula solves any cubic, but it is long and can require complex numbers even when all roots are real. The calculator picks the cleanest method.
Can a cubic have no real roots?
No. A real cubic always crosses the x-axis at least once.
What if a = 0?
Then the equation is quadratic. Use the quadratic formula calculator.
Why are some roots shown as decimals?
When no root is rational, exact forms involve cube roots and are usually messy, so the calculator gives accurate decimals.
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