Cubic Equation Solver

1. Write your equation as a x^3 + b x^2 + c x + d = 0.
2. Enter the four coefficients a, b, c and d.
3. Read the roots, whether real or complex.
4. See how many are real and the nature of the solution.

Here is what each result means:

A cubic equation is a third-degree equation of the form a x^3 + b x^2 + c x + d = 0, where a is not zero. Its graph is a curve that can bend twice, and it always crosses the x-axis at least once. So a cubic has one or three real roots, and any roots that are not real come as a complex pair.

Cubics appear in geometry, physics and engineering, wherever a volume, a curve or a balance leads to a third-degree relation. Solving one by hand is much harder than a quadratic, which is why a solver that handles every case is so useful.

It divides through by the leading coefficient, then applies the standard cubic method to find all three roots.

1. Normalize the equation so the leading coefficient is one.
2. Form the discriminant to decide the type of roots.
3. Use radicals for one real root, or a trigonometric form for three real roots.

If the leading coefficient is zero, the equation is really a quadratic; the quadratic equation calculator handles that case.

Solve x^3 - 6x^2 + 11x - 6 = 0.

Calculation: this cubic factors as (x - 1)(x - 2)(x - 3), so the three real roots are 1, 2 and 3. The solver reaches the same answer directly from the coefficients, without needing you to spot the factors.

The discriminant decides the mix of real and complex roots.

Whatever the case, a cubic always has at least one real root, because its curve runs from far below to far above the axis.

A cubic is a step up in degree, and in difficulty, from a quadratic.

Unlike a quadratic, a cubic can never have zero real roots, since its curve must cross the axis at least once.

The Discriminant

Its sign sets whether you get one real root with a complex pair, or three real roots.

The Leading Coefficient

It stretches the curve and sets which way the ends point, but a of zero makes it a quadratic.

The Constant Term

It shifts the curve up or down, moving where it crosses the x-axis.

Algebra Homework

Find roots that do not factor neatly, including complex ones.

Geometry and Volume

Solve a third-degree relation for a length when a volume is fixed.

Physics and Engineering

Handle cubic relations that arise in equations of state or stability.

1. Missing a Root

A cubic has three roots counting repeats. Do not stop at the first one you find.

2. Ignoring Complex Roots

When only one root is real, the other two are complex, not nonexistent.

3. Treating It as a Quadratic

If a is not zero, it is a cubic. Only a of zero drops it to a quadratic.

4. Sign Errors in Coefficients

Move every term to one side first and read the signs carefully.

5. Rounding Too Soon

Small rounding can hide a neat whole-number root. Keep precision and round at the end.

The method is exact; only the displayed decimals are rounded.

Write it as a x cubed plus b x squared plus c x plus d equals zero, then apply the cubic method. This tool normalizes the equation and returns all three roots, real or complex, from the coefficients.

Exactly three, counting repeats. Either all three are real, or one is real and the other two are a complex pair. A cubic always has at least one real root.

No. Unlike a quadratic, a cubic curve runs from far below the axis to far above it, so it must cross at least once. There is always at least one real root.

When only one root is real, the other two are complex numbers that form a conjugate pair, with equal real parts and opposite imaginary parts. This happens when the discriminant is positive.

Then it is not a cubic. With a of zero the highest power is x squared, so it is a quadratic. Use a quadratic solver, which this tool points you to.

Yes, but it is far more complex. This solver uses the standard cubic method, including a trigonometric form for the case of three real roots, so you do not have to work through it by hand.

Results are rounded to four decimals for display. A root that is really a whole number can show a tiny rounding difference in the last place.

A cubic has x cubed as its highest power and exactly three roots counting repeats, while a quadratic tops out at x squared with up to two roots. A cubic always has at least one real root.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Cubic equation (Wikipedia).
• Solving cubic equations (Maths Is Fun).
• Cubic function (Wikipedia).