Solve any cubic equation ax³ + bx² + cx + d = 0 instantly. Enter the four coefficients to find all real roots with a clear step-by-step breakdown — free, no sign-up, works on any device.
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A cubic equation is a polynomial equation of degree three, written in the general form ax³ + bx² + cx + d = 0, where a is not zero. Cubics sit at an interesting crossroads in mathematics: they are the highest-degree polynomial equations that still have a general solution formula, yet that formula — discovered in 16th-century Italy — is so intricate that for centuries mathematicians preferred clever factoring tricks instead. This free cubic equation calculator removes all of that friction. Type in the four coefficients and it instantly returns every real root, shows the discriminant, and walks you through the classical solution steps so you can follow the reasoning, not just copy an answer.
Unlike a quadratic, which has at most two real solutions, a cubic always has at least one real root — its graph must cross the x-axis somewhere because it stretches from negative infinity on one side to positive infinity on the other. Depending on the shape of the curve, you can get one simple real root, a repeated (double) root plus a simple root, or three distinct real roots. The calculator detects which case applies automatically and even handles the degenerate situations gracefully: if you enter a = 0 it solves the resulting quadratic or linear equation instead of throwing an error.
Using the solver takes less than a minute:
A quick tip for students: after the calculator gives you a root, substitute it back into the original equation to verify it equals zero. This one habit catches nearly every sign error, and the worked examples below show exactly how the check is done.
A cubic equation is any equation that can be rearranged into the form ax³ + bx² + cx + d = 0 with a ≠ 0. The word “cubic” comes from the geometric cube: just as x² measures the area of a square with side x, x³ measures the volume of a cube with side x. That is why cubics appear naturally whenever volume, three-dimensional growth, or triple products are involved — from sizing a shipping container to modeling how a population compounds over three interacting factors.
Every cubic has exactly three roots if you count complex numbers and multiplicity (this is the Fundamental Theorem of Algebra at work). What varies is how many of those roots are real numbers you can plot on a number line:
The discriminant of the cubic tells you which case you are in before computing a single root. For the depressed cubic t³ + pt + q = 0 it is Δ = (q/2)² + (p/3)³: a positive discriminant means one real root, zero means a repeated root, and a negative discriminant means three distinct real roots. The calculator evaluates this number for you and explains what it implies — a genuinely useful piece of mathematical insight that textbooks often bury.
The general solution to the cubic was first published by Gerolamo Cardano in his 1545 book Ars Magna, based on discoveries by Scipione del Ferro and Nicolò Tartaglia. The method works in two stages. First, the cubic is depressed — the quadratic term is eliminated with the substitution x = t − b/3a, producing the simpler equation t³ + pt + q = 0, where:
p = c/a − b²/3a²
q = 2b³/27a³ − bc/3a² + d/a
Second, Cardano’s formula gives a root of the depressed cubic directly:
t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ)
then shift back with x = t − b/3a
There is a famous subtlety called the casus irreducibilis (“the irreducible case”): when the discriminant is negative and all three roots are real, Cardano’s formula forces you to take cube roots of complex numbers — real answers emerge only after the imaginary parts cancel. Sixteenth-century mathematicians found this deeply disturbing, and it was one of the historical pressures that led to the acceptance of complex numbers. Modern calculators sidestep the paradox with the trigonometric form: tk = 2√(−p/3)·cos(φ/3 − 2kπ/3), which produces the three real roots using only real arithmetic. That is exactly what this calculator does behind the scenes whenever Δ < 0.
Example 1 — three distinct real roots. Solve x³ − 6x² + 11x − 6 = 0. The coefficients are a = 1, b = −6, c = 11, d = −6. Dividing by a changes nothing. Depress with x = t + 2 (since −b/3a = 2): expanding (t + 2)³ − 6(t + 2)² + 11(t + 2) − 6 gives t³ − t = 0, so p = −1 and q = 0. The discriminant is Δ = 0 + (−1/3)³ = −1/27, which is negative — three real roots. The trigonometric form yields t = −1, 0, 1, and shifting back by 2 gives x = 1, 2, 3. Check: (1)³ − 6(1)² + 11(1) − 6 = 1 − 6 + 11 − 6 = 0. This cubic factors neatly as (x − 1)(x − 2)(x − 3) = 0.
Example 2 — one real root. Solve x³ + x + 1 = 0. Here a = 1, b = 0, c = 1, d = 1, and the cubic is already depressed (p = 1, q = 1). The discriminant is Δ = (1/2)² + (1/3)³ = 1/4 + 1/27 ≈ 0.287, positive, so there is exactly one real root. Cardano’s formula gives t = ∛(−0.5 + √0.287) + ∛(−0.5 − √0.287) ≈ ∛(0.0358) + ∛(−1.0358) ≈ 0.329 − 1.011 = −0.682. So x ≈ −0.682, with the remaining two roots complex. A quick check: (−0.682)³ + (−0.682) + 1 ≈ −0.317 − 0.682 + 1 ≈ 0.
Example 3 — a repeated root. Solve x³ − 3x² + 3x − 1 = 0. Notice this is the expansion of (x − 1)³. The discriminant works out to exactly zero, and the calculator reports the triple root x = 1 (multiplicity 3). Whenever you spot the pattern of binomial coefficients 1, 3, 3, 1 with alternating signs, you are looking at a perfect cube — factoring by recognition is far faster than the formula.
Example 4 — leading coefficient not equal to 1. Solve 2x³ − 4x² − 22x + 24 = 0. Step 1 is always to divide everything by a = 2: x³ − 2x² − 11x + 12 = 0. Trying integer factors of 12, x = 1 gives 1 − 2 − 11 + 12 = 0, so (x − 1) is a factor. Polynomial division leaves x² − x − 12 = (x − 4)(x + 3), so the roots are x = 1, 4, −3. The calculator reaches the same answer through the depressed cubic, but the Rational Root Theorem — test factors of d/a against factors of the leading coefficient — often cracks textbook cubics in seconds.
Box and container design. A manufacturer wants an open-top box with volume 1,000 cm³, cut from a square sheet by removing x-by-x corners. If the sheet is 40 cm wide, the volume equation is x(40 − 2x)² = 1000, which expands to a cubic in x. Solving it tells the designer exactly how large to cut the corners — a calculation done thousands of times a day in packaging engineering.
Finance and compound growth. An investment that grows through three successive multiplicative stages — say, a currency conversion, then interest, then a fee multiplier — leads to equations where the unknown appears cubed. Economists also meet cubics in certain supply-demand equilibrium models where price enters through cubic cost functions.
Physics and engineering. The van der Waals equation for real gases is cubic in volume; solving it predicts how a gas compresses under pressure far more accurately than the ideal gas law. In optics, the angle of minimum deviation through a prism involves cubic relationships, and structural engineers encounter cubics when analyzing beam deflections under certain load patterns.
Computer graphics. Bézier curves — the mathematical backbone of fonts, vector illustration, and animation paths — are cubic polynomials. Finding where a Bézier curve crosses a line, or splitting a curve at a given point, reduces to solving a cubic equation, executed millions of times per second by your GPU.
Keep exploring polynomial mathematics with these free tools: the Quadratic Formula Calculator solves second-degree equations with the familiar discriminant method; the Exponent Calculator handles the power computations that appear inside Cardano’s formula; the Scientific Calculator covers trigonometric and root functions for hand-checking the casus irreducibilis; and the Fraction Calculator keeps your rational-root test arithmetic exact.
This page provides a free cubic equation calculator that solves ax³ + bx² + cx + d = 0 for all real roots using Cardano’s method with the trigonometric form for the casus irreducibilis. It includes the depressed-cubic derivation, discriminant classification (Δ > 0: one real root; Δ = 0: repeated root; Δ < 0: three real roots), worked numeric examples, real-world applications, and common-mistake guidance. Canonical URL: https://sigmacalculator.net/math-calculators/cubic-equation-calculator/.
A cubic equation is a polynomial equation of degree three, written in standard form as ax³ + bx² + cx + d = 0 where a is not zero. The highest power of the variable is 3, which is why it is called “cubic” — just as x² relates to the area of a square, x³ relates to the volume of a cube.
First divide every term by a so the equation starts with x³. Next, depress the cubic with the substitution x = t − b/3a to get t³ + pt + q = 0. Then compute the discriminant Δ = (q/2)² + (p/3)³. Finally, apply Cardano’s formula — or the trigonometric form when Δ is negative — and shift back with x = t − b/3a. The calculator above performs all four steps automatically and shows its work.
No. A cubic equation with real coefficients always has at least one real root, because its graph stretches from negative infinity on one end to positive infinity on the other and must cross the x-axis somewhere. It can have one real root (plus two complex ones), a repeated root, or three distinct real roots — but never zero real roots.
For the depressed cubic t³ + pt + q = 0, the discriminant is Δ = (q/2)² + (p/3)³. If Δ is positive there is one real root and a complex pair; if it is zero there is a multiple (repeated) root; if it is negative there are three distinct real roots. It plays the same classifying role as b² − 4ac does for quadratics.
Cardano’s formula gives a root of the depressed cubic t³ + pt + q = 0 as t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ), where Δ is the discriminant. Published by Gerolamo Cardano in 1545 (based on work by del Ferro and Tartaglia), it is the cubic analogue of the quadratic formula. When Δ is negative, the trigonometric form is used instead to avoid complex intermediate numbers.
If a = 0, the equation is no longer cubic — it collapses to the quadratic bx² + cx + d = 0 (or to a linear equation if b is also zero). This calculator detects that automatically and solves the lower-degree equation instead, so you still get the correct roots without an error.
Substitute each root back into the original equation — the result should be zero (up to tiny rounding error). You can also multiply out the factors: if the roots are r₁, r₂, r₃, then a(x − r₁)(x − r₂)(x − r₃) should expand back to your original polynomial. Both checks take seconds and catch nearly every sign mistake.
Cubics appear in packaging design (box volume problems), the van der Waals equation for real gases, compound growth models in finance, beam deflection in structural engineering, and computer graphics — Bézier curves used in fonts and animation are cubic polynomials, and finding where they intersect lines requires solving cubics millions of times per second.
Cite this page as: Cubic Equation Calculator. Sigma Calculator. https://sigmacalculator.net/math-calculators/cubic-equation-calculator/.
Last updated: October 9, 2026.