Cubic Equation Calculator
Every cubic has three roots counting multiplicity: x³ − 6x² + 11x − 6 = 0 has x = 3, x = 2, x = 1.
x = 3, x = 2, x = 1
- Sum of roots −b/a
- 6
- Product of roots −d/a
- 6
- Real roots
- 3
Step-by-step solution
- Solve ax³ + bx² + cx + d = 0 (Cardano / trigonometric method)
- Check: the roots add up to −b/a = 6 and multiply to −d/a = 6
- x = 3, x = 2, x = 1
Cubic Equation Formula
ax³ + bx² + cx + d = 0
roots sum to −b/a and multiply to −d/a
Worked Example
The equation x³ − 6x² + 11x − 6 = 0:
- Solve ax³ + bx² + cx + d = 0 (Cardano / trigonometric method)
- Check: the roots add up to −b/a = 6 and multiply to −d/a = 6
- x = 3, x = 2, x = 1
Product of roots −d/a: 6 · Real roots: 3
Test-day tip
Polynomial questions on the SAT and ACT often give the factored form; the x-intercepts are the values that make each factor zero. A factor squared, (x − 2)², is a root where the graph touches the axis without crossing.
More Advanced Math Calculators
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Quadratic Formula Calculator
x = [−b ± √(b² − 4ac)] ÷ 2a
discriminant b² − 4ac: > 0 two real roots, = 0 one, < 0 complex
Example: The equation x² − 5x + 6 = 0: Roots: x = 2, x = 3.
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Vertex Form Calculator
y = a(x − h)² + k, h = −b/2a, k = c − ah²
vertex (h, k); a > 0 opens up, a < 0 down
Example: The parabola y = 2x² − 8x + 5: Vertex form: y = 2(x − 2)² − 3.
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Exponential Growth Calculator
y = a(1 ± r)ᵗ
a initial value, r rate per period, t periods; continuous: y = a·e^(±rt)
Example: 500 growing 8% per period for 5 periods: Value after t periods = 734.6640384.
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Simplify Radicals Calculator
√(k²·m) = k√m
take out the largest perfect square (or cube) factor
Example: The square root of 72: Simplified = 6√2.
Frequently asked questions
How can I find one root of a cubic by hand?
Try factors of d over factors of a (the rational root theorem). If x = 1 works, divide by (x − 1) and solve the remaining quadratic.
Can a cubic have no real roots?
No. A cubic always crosses the x-axis at least once, so it has one or three real roots.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
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APA
Scientific Converters. (2026, October 15). Cubic Equation Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/cubic-equation-calculator/
IEEE
Scientific Converters, "Cubic Equation Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/cubic-equation-calculator/