Math
Cubic Equation Calculator
Enter the four coefficients or type the equation and the cubic equation calculator finds all three roots.
When a rational root exists it is found exactly with the rational root theorem, divided out with synthetic division, and the remaining quadratic is solved with exact radicals or complex a ± bi. Otherwise the depressed cubic is solved with Cardano's formula or the trigonometric method to 10-digit decimals. You also get the discriminant, factored form, turning points, inflection point, and a graph, or work backward and build a cubic from three roots.
Cannot be 0.
Use 0 if missing.
Use 0 if missing.
Use 0 if missing.
Try an example
Result
Roots
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- Roots (decimal)
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- Discriminant
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- Type of roots
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- Factored form
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- Depressed cubic
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- Standard form
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More details (1 more)
- Turning points
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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What the cubic equation calculator solves
A cubic equation ax^3 + bx^2 + cx + d = 0 always has exactly three roots counted with multiplicity: either three real roots or one real root plus a pair of complex conjugates. This page finds them in the form a student is expected to write: whole numbers and fractions when the polynomial factors, simplified A root expression written with the √ symbol, such as a square root. when a quadratic factor remains, and decimals only when no closed form is reasonable.
How the roots are found
- Clear fractions and decimals so the The number multiplying a variable, like the 3 in 3x. are integers (the roots do not change).
- List the rational root candidates ±p/q, where p divides the constant term and q divides the leading coefficient, and test each one exactly.
- If one is a root, divide it out with synthetic division. The quotient is a quadratic, which the quadratic formula solves exactly.
- If no candidate works, shift to the depressed cubic t^3 + pt + q = 0. Δ > 0 means three real roots (trigonometric method); Δ < 0 means one real root (Cardano's formula).
- Polish every decimal root with Newton's method and check f(root) ≈ 0.
Writing a cubic from its roots
Every root r gives a factor (x - r). Multiply the three factors, then multiply by any leading coefficient you want. Roots 1, 2, 3 give (x - 1)(x - 2)(x - 3) = x^3 - 6x^2 + 11x - 6. Choose Build a cubic from three roots to see each multiplication written out; repeat a root to create a double root that only touches the x-axis.
Reading a cubic from its graph
- Each x-A point where a graph crosses an axis. is a real root. A crossing is a single root; a touch-and-turn is a double root; a flat crossing through the axis is a triple root.
- The y-intercept is d, the constant term.
- The ends tell you the sign of a: rising to the right means a > 0, falling means a < 0.
- Turning points (local max and min) sit where 3ax^2 + 2bx + c = 0; the inflection point at x = -b/(3a) is halfway between them and is the graph's center of symmetry.
Reading the discriminant
| Δ | Roots | Graph |
|---|---|---|
| Δ > 0 | Three distinct real roots | Crosses the x-axis three times |
| Δ = 0 | A repeated real root (double or triple) | Touches the x-axis at the repeated root |
| Δ < 0 | One real root and two complex conjugates | Crosses the x-axis once |
Common mistakes
- Skipping the rational root test and jumping to Cardano's formula, which hides simple integer roots inside nested radicals.
- Forgetting negative candidates and fractions ±p/q when the leading coefficient is not 1.
- Stopping after one root. Always divide it out and solve the quadratic that remains.
- Entering a = 0. That is a quadratic, not a cubic.
Worked examples
Three integer roots
x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)
Roots
x = 1, x = 2, x = 3
Rational root plus complex pair
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
Roots
x = 2, x = -1 ± (√3)i
Fractional root and radicals
2x^3 - x^2 - 6x + 3 = (2x - 1)(x^2 - 3)
Roots
x = 1/2, x = -√3, x = √3
No rational root, three real roots
x^3 - 3x + 1 = 0 solved with the trigonometric method
Roots
x ≈ -1.879385, x ≈ 0.347296, x ≈ 1.532089
No rational root, one real root
x^3 + x + 1 = 0 solved with Cardano's formula
Roots
x ≈ -0.682328, x ≈ 0.341164 ± 1.161541i
Triple root
x^3 - 3x^2 + 3x - 1 = (x - 1)^3
Roots
x = 1 (triple root)
Typed equation
x^3 - 7x + 6 = 0 has roots -3, 1, 2
Roots
x = -3, x = 1, x = 2
Double root at the origin's neighbour
x^3 - 4x^2 + 4x = x(x - 2)^2, so x = 2 is a double root
Roots
x = 0, x = 2 (double root)
Negative rational root with a complex pair
x^3 + 2x^2 + 3x + 2 = (x + 1)(x^2 + x + 2)
Roots
x = -1, x = -1/2 ± (√7/2)i
Build from roots
Roots 1, 2, 3 give x^3 - 6x^2 + 11x - 6 = 0
Roots
x = 1, x = 2, x = 3
Build from fractional roots with a leading coefficient
Roots 1/2, -1, 3 with a = 2 give 2x^3 - 5x^2 - 4x + 3 = 0
Roots
x = -1, x = 1/2, x = 3
a = 0 is not a cubic
0x^3 + x^2 - 1 = 0 is quadratic
Roots
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Frequently asked questions
How many roots does a cubic equation have?+
Always three, counting multiplicity. Either all three are real, or one is real and the other two are complex conjugates. A cubic always has at least one real root.
What is the rational root theorem?+
If a polynomial with integer coefficients has a rational root p/q in lowest terms, then p divides the constant term and q divides the leading coefficient. That gives a short list of candidates to test.
When does the calculator give decimals instead of exact roots?+
Only when no rational root exists. Then the exact form would be Cardano's nested radicals or a cosine expression, which are rarely useful in class, so the calculator shows 10-digit decimals and the depressed cubic used to get them.
What is a depressed cubic?+
A cubic with no x^2 term, t^3 + pt + q = 0. Any cubic becomes depressed with the substitution x = t - b/(3a). Cardano's formula and the trigonometric method both start from this form.
What does the discriminant of a cubic tell me?+
Its sign gives the number of real roots: positive means three distinct real roots, zero means a repeated root, negative means exactly one real root.
Can I type the equation with terms on both sides?+
Yes. Choose Type the equation and enter it as written, such as 2x³ + 3x² = 11x + 6. Everything is moved to one side before solving.
How do I find a cubic equation when the roots are given?+
Turn each root r into a factor (x - r) and multiply: roots -1, 2, and 5 give (x + 1)(x - 2)(x - 5) = x^3 - 6x^2 + 3x + 10. Multiply by a leading coefficient if the problem specifies one. The Build a cubic from three roots mode does the expansion step by step.
How do I write a cubic equation from its graph?+
Read the x-intercepts as roots (a touch without crossing is a double root), build the factors, and fix the leading coefficient with one more point, usually the y-intercept: if the graph crosses the y-axis at d, then a × (product of the negated roots) must equal d.
How does synthetic division help solve a cubic?+
Once one root r is known, synthetic division by (x - r) reduces the cubic to a quadratic in three quick rows: bring down the leading coefficient, multiply by r, add, repeat. The remainder must be 0; the other numbers are the quadratic's coefficients, which the quadratic formula finishes.
How do I factor a cubic polynomial?+
First pull out any common factor (including x when there is no constant term). Then test rational root candidates ±p/q; each root r you find gives a factor (x - r). Divide it out and factor the remaining quadratic if its discriminant is a perfect square, otherwise leave it as an irreducible quadratic factor.
What are the turning points of a cubic?+
The local maximum and minimum, where the derivative 3ax^2 + 2bx + c equals 0. A cubic has two turning points when b^2 - 3ac > 0, one flat point when it equals 0, and none when it is negative, in which case the curve rises or falls the whole way.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026