About the Cubic Equation Solver
A cubic equation has an x cubed term, and it can have one, two or three real solutions. Unlike the quadratic there is no short formula that most people remember, so solving by hand is slow. This solver takes the four coefficients and gives all three roots, including complex ones written with an i, and tells you which case it is: three different real roots, a repeated root or one real root and a complex pair. Type the four coefficients a, b, c and d and the roots of a cubic equation appears at once, with a few related figures underneath, and you can copy the result with one tap. The sums run in your browser, so nothing you enter is uploaded.
Updated October 2026
How to use the Cubic Equation Solver
- 1Enter the four coefficients a, b, c and d.
- 2Pick the units or options from the menus if the roots of a cubic equation tool offers them.
- 3Read the roots of a cubic equation in the large result box.
- 4Look at the extra rows for related figures.
- 5Press Copy result to keep the answer.
When you need the roots of a cubic equation
Students use cubic solvers to check homework on polynomials and to see the graph behaviour: three real roots mean the curve crosses the axis three times. Engineers meet cubics in equations of state for gases, in beam deflection and in some control problems.
The coefficient a must not be zero; otherwise the equation is a quadratic, which has its own solver. Dividing the whole equation by a does not change the roots, so you can use any scale. If all roots are real, the complex part shows as 0.
How the roots of a cubic equation is worked out
The equation a·x³ + b·x² + c·x + d = 0 is first reduced to t³ + p·t + q = 0 with t = x + b ÷ 3a. The sign of the discriminant −(4p³ + 27q²) says how many roots are real, and Cardano's method or a trigonometric form gives them. The equation x³ − 6x² + 11x − 6 = 0 has three real roots, 1, 2 and 3, because it factors as (x − 1)(x − 2)(x − 3). The equation x³ − 1 = 0 has one real root, 1, and two complex roots, −0.5 ± 0.866i. The equation x³ − 3x + 2 = 0 has a repeated root: 1 twice and −2.
Every answer on this page comes from that one rule, so you can check any result by hand with a calculator. The figures shown underneath, such as other units, are conversions of the same answer.
- Real and complex roots
- Says which case it is
- Handles repeated roots
Mistakes to avoid with roots of a cubic equation
Complex roots always come in pairs for an equation with real coefficients: if −0.5 + 0.866i is a root, so is −0.5 − 0.866i. Rounding to six significant figures can make a repeated root show tiny differences, so look at the kind of roots reported rather than at the last digit.
On phone and computer
On a phone, type into the boxes one after another and read the roots of a cubic equation underneath; Copy result puts it on the clipboard for a message or a note. On a computer the inputs sit on the left and the answer on the right, so you can change one figure and watch the roots of a cubic equation move. The cubic equation solver works the same on both. For simpler equations, Quadratic Equation Solver and Linear Equations Solver are faster. Derivative Calculator finds turning points, and Complex Number Calculator helps when roots are not real.
Features
- a cannot be 0
- Complex roots come in pairs
- Check by substituting back
- Clear messages when the roots of a cubic equation inputs make no sense
- Copy the result with one tap
- Private: nothing is uploaded
Frequently asked questions
How do I calculate roots of a cubic equation?
The equation a·x³ + b·x² + c·x + d = 0 is first reduced to t³ + p·t + q = 0 with t = x + b ÷ 3a. The sign of the discriminant −(4p³ + 27q²) says how many roots are real, and Cardano's method or a trigonometric form gives them. The equation x³ − 6x² + 11x − 6 = 0 has three real roots, 1, 2 and 3, because it factors as (x − 1)(x − 2)(x − 3). The equation x³ − 1 = 0 has one real root, 1, and two complex roots, −0.5 ± 0.866i. The equation x³ − 3x + 2 = 0 has a repeated root: 1 twice and −2.
What is the formula for roots of a cubic equation?
The equation a·x³ + b·x² + c·x + d = 0 is first reduced to t³ + p·t + q = 0 with t = x + b ÷ 3a. The sign of the discriminant −(4p³ + 27q²) says how many roots are real, and Cardano's method or a trigonometric form gives them. The cubic equation solver applies it exactly and rounds only the displayed answer to six significant figures.
How many roots does a cubic have?
Always three when counted with multiplicity, real or complex. At least one is always real.
What is the discriminant?
A number that tells how the roots behave. Positive means three different real roots, zero a repeated root, negative one real root and two complex ones.
How do I check a root?
Put it back into the equation. The result should be zero, apart from tiny rounding error.
Is this cubic equation solver free and private?
Yes. It is free, needs no account and runs in your browser, so the numbers you type are not uploaded or stored.