Cubic Equation Solver
Enter a, b, c and d from ax³ + bx² + cx + d = 0. You get the real and complex roots and the discriminant, with exact answers where the roots are whole numbers or fractions.
If a term is missing, enter 0 for it. The number a cannot be zero.
Worked examples
Three whole-number roots
x³ - 6x² + 11x - 6 = 0
x = 1, 2 and 3
Factorises as (x - 1)(x - 2)(x - 3).
One real root
x³ - 1 = 0
x = 1, and -0.5 ± 0.866i
The complex roots are the other cube roots of 1.
About Cubic Equation Solver
A cubic equation has x³ as its highest power, and it always has three roots once complex numbers are counted. Either all three roots are real, or there is one real root and a pair of complex ones that mirror each other. The discriminant tells you which: positive means three different real roots, zero means a repeated root, and negative means one real root with two complex roots. A cubic curve always crosses the x axis at least once.
The solver first looks for a root that is a whole number or a fraction, using the rational root theorem, which says such a root must be a factor of d divided by a factor of a. If it finds one, it divides it out and solves the remaining quadratic exactly. For x³ - 6x² + 11x - 6 = 0 that gives 1, 2 and 3. When there is no simple root, it uses the closed formulas for a cubic and refines the answer, so the decimals are accurate to many places. The discriminant is calculated exactly so the classification is always right.
How to use Cubic Equation Solver
- Write your equation as ax³ + bx² + cx + d = 0.
- Enter a, b, c and d, using 0 for any missing term. The number a cannot be 0.
- Read the real roots and any complex pair.
- Look at the table for the discriminant and any exact root that was found.
Common uses
- Solving cubic equations in A level and university maths
- Finding the dimensions of a box from a volume equation
- Engineering formulas that lead to a cubic
- Checking roots found by factorising
Frequently asked questions
How many roots does a cubic have?
Three, counting repeated roots and complex roots. At least one is always real.
What if my equation has no x³ term?
Then it is a quadratic, so use the quadratic equation solver. The number a must not be zero here.
What is the rational root theorem?
It says any fraction p/q that is a root must have p dividing the constant term and q dividing the x³ number. Trying those few candidates is a quick way to find a first root.
What does a repeated root mean?
The curve touches the x axis and turns back instead of crossing it. For x³ - 3x² + 3x - 1, the root 1 appears three times.
Are the decimal answers exact?
They are accurate to many decimal places. When a root is a whole number or a fraction, it is found exactly and shown in that form too.