Cubic Equation Calculator - Real, Complex, Repeated Roots

A cubic equation calculator solves any polynomial equation of the form ax³+bx²+cx+d=0 and returns all three roots—real, repeated, or complex conjugate—along with the cubic discriminant and a plain-language solution type. This solver handles every case that the Cardano closed-form formula can describe, including repeated roots, three distinct real roots, and the casus irreducibilis where intermediate cube roots go through the complex numbers even though the final answers are real.

• • Algebra and precalculus homework: Verify the roots of ax³+bx²+cx+d=0 quickly while learning Cardano formula in class.
• • Engineering root-finding: Find the volume or load roots that drive a cubic design equation in mechanical, civil, or chemical engineering.
• • Curve sketching: Locate the x-intercepts of a cubic polynomial to sketch or analyze its graph.
• • Numerical-method benchmarking: Compare Newton's method, Vieta substitution, and direct Cardano method against a closed-form reference.

A cubic equation is a degree-3 polynomial, and the fundamental theorem of algebra means there are exactly three roots counting multiplicity, but those roots can be real, repeated, or appear as a complex conjugate pair. The solver applies Cardano formula so you do not need to memorize the depressed-cubic substitution or the trigonometric form needed for the casus irreducibilis. The tool also reports the cubic discriminant, which is the cleanest single number that tells you whether to expect three real roots, one real root plus two complex ones, or a repeated root.

Use this solver any time you have ax³+bx²+cx+d=0 with real coefficients and want every root without doing the algebra by hand. The default sample values (1, -6, 11, -6) are the canonical (x-1)(x-2)(x-3) example, so the result panel shows x = 1, 2, 3 the first time the page loads.

For the lower-degree cousin of this problem, the Quadratic Formula Calculator covers the discriminant and vertex for ax squared plus bx plus c equals 0 in the same result-panel format.

The solver normalizes the equation so the leading coefficient is 1, applies the depressed-cubic substitution t = x + b/(3a) to remove the x² term, and then solves the resulting t³+Pt+Q=0 with Cardano formula. The trigonometric branch is used when the depressed-cubic discriminant is positive and the casus irreducibilis would otherwise force complex intermediate arithmetic.

• a: Leading coefficient of x cubed. Must be non-zero; the solver rejects a = 0 with a validation error.
• b: Coefficient of x squared. After normalization it sets the shift used in the depressed-cubic substitution.
• c: Coefficient of x. Combined with the normalized b and d to build the depressed-cubic coefficient P.
• d: Constant term. Combined with the normalized b and c to build the depressed-cubic coefficient Q.

The depressed-cubic substitution eliminates the x squared term and turns the problem into a simpler equation with only t and t cubed terms. Once the calculator recovers the three t-roots, it subtracts the shift p/3 to map them back to the original x-variable. According to Wolfram MathWorld, this depressed-then-solve pipeline is exactly the procedure Gerolamo Cardano published in Ars Magna in 1545, and it is the fastest closed-form way to get all three roots at once.

According to Wolfram MathWorld, a general cubic ax^3+bx^2+cx+d=0 is reduced to the depressed cubic t^3+Pt+Q=0 by substituting x = t - b/(3a) and is then solved with Cardano's closed-form expression involving cube roots of Q/2 plus or minus the square root of the discriminant.

When you already know one root r of the cubic, divide ax cubed plus bx squared plus cx plus d by (x minus r) on the Polynomial Division Calculator to reduce the problem to a quadratic you can finish by inspection.

Four ideas drive this solver: Cardano formula, the depressed-cubic substitution, the cubic discriminant, and the casus irreducibilis. Mastering these terms makes the result panel easy to interpret.

If the cubic equation calculator returns a+bi and a-bi roots, the Complex Number Calculator is the right follow-up for arithmetic with those complex conjugates.

Enter the four coefficients, watch the result panel update, and copy the formatted roots into your homework or design notes. The default sample is the canonical (x-1)(x-2)(x-3) cubic so you can see a finished result immediately.

1. 1 Enter the coefficient of x cubed: Type the value of a into the first field. The solver rejects a = 0 with an inline error because a degree-3 equation needs a non-zero leading coefficient.
2. 2 Enter the coefficient of x squared: Type the value of b. This is the term the depressed-cubic substitution will eliminate with the shift p/3.
3. 3 Enter the coefficient of x: Type the value of c. The calculator uses it to build the depressed-cubic coefficient P and the cubic discriminant.
4. 4 Enter the constant term: Type the value of d. The product of the three roots is -d/a, so the solver will recover values that match this identity to four decimal places.
5. 5 Read the roots and discriminant: The result panel shows x₁, x₂, x₃, the cubic discriminant, and a plain-language solution type. Complex roots appear as a+bi / a-bi strings.
6. 6 Use Reset to start over: Press Reset to restore the (1, -6, 11, -6) defaults or to clear a faulty input before trying a new equation.

Practical usage: a student verifying x³ - 6x² + 11x - 6 = 0 types 1, -6, 11, -6, sees 1, 2, 3 in the result panel, and confirms the discriminant of 4 against their textbook entry. An engineer solving 2x³ - 12x² + 22x - 12 = 0 leaves the same numbers with a = 2 and the solver still reports 1, 2, 3, demonstrating that scaling the polynomial does not change the roots.

After you have the three real roots, plot the cubic on the Polynomial Graphing Calculator to see each root land on the x-axis and confirm the result visually.

This tool saves time on the algebra, surfaces the discriminant and solution type, and makes the casus irreducibilis case just as easy to read as the ordinary three-real-root case.

• • All three roots in one pass: Cardano formula returns every root at once, so this tool saves you from doing long division, synthetic division, or repeated guess-and-check by hand.
• • Discriminant built in: The cubic discriminant is reported as a numeric field, so you do not have to remember the 18abcd-4b³d+b²c²-4ac³-27a²d² formula to know whether to expect real or complex answers.
• • Casus irreducibilis handled automatically: When the depressed-cubic discriminant is positive and the trigonometric form is more stable, the calculator switches branches so the result panel never returns complex garbage for a real problem.
• • Repeated-root detection: A near-zero discriminant triggers the double/triple-root branch, so the solver reports the multiplicity of each root instead of silently merging them.
• • Mobile-friendly output: The result panel is two-column on tablet and desktop, one-column on phones, and the Submit button scrolls the mobile view to the results.
• • Verifiable by hand: Every value comes from Cardano formula and Vieta formulas, so this tool is a reliable reference for textbook problems and engineering design checks.

When the cubic is part of a larger system where one equation is linear in x, the System of Equations Calculator can solve the linear piece while the cubic equation calculator handles the nonlinear one.

This tool is built on Cardano closed-form formula, so the results depend on the leading coefficient being non-zero, on the discriminant sign, and on the precision used for the trigonometric branch.

• • This tool handles real coefficients only. If you need a cubic with complex coefficients you will need a more general polynomial solver.
• • Results are rounded to 4 decimal places. If your work demands more precision, recompute Cardano formula with higher-precision arithmetic and compare against the panel.

According to Wikipedia, the cubic discriminant Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2 is positive when three distinct real roots exist, zero when there is a multiple root, and negative when exactly one real root and two non-real complex conjugate roots exist.