Free Cubic Equation Calculator

What Is a Cubic Equation?

A cubic equation is just a polynomial where the highest power of x is 3. Written out in standard form, it looks like this, where a can never be 0 (otherwise the x³ term disappears and it's no longer a cubic):

ax³ + bx² + cx + d = 0

The b, c and d terms are all allowed to be missing (or zero) — only the a term is required.

Worked Example

Solve 22x³ + 25x² + 14x + 5 = 0:

x₁ = −0.6914
x₂ = −0.22248 + 0.52841i
x₃ = −0.22248 − 0.52841i

Here there's one real solution and a pair of complex solutions, which is a completely normal outcome for a cubic — not every cubic factors into three neat real numbers.

How This Calculator Solves It

Behind the scenes, this uses Cardano's method: the equation first gets simplified into a "depressed" cubic with no x² term, then a substitution turns it into something that can be solved with cube roots directly. Depending on a value called the discriminant, the result comes out as either three real roots or one real root plus a complex conjugate pair, exactly like the quadratic formula's discriminant tells you whether you're dealing with real or complex roots — just one degree up.

A Shortcut When There's No Constant Term

If d = 0, you can skip straight to factoring out an x, since x = 0 is automatically one of the solutions. What's left is a plain quadratic, which you can solve with the quadratic formula, completing the square, or factoring — whichever's fastest for that equation.

Vieta's Formulas: A Useful Shortcut

If you already know one root of a cubic, Vieta's formulas let you find the other two without repeating the full method. For roots p, q and r of ax³+bx²+cx+d=0:

p + q + r = −b/a
pq + qr + rp = c/a
pqr = −d/a

Since these are just three equations, knowing any one root lets you solve for the remaining two using ordinary algebra.

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Frequently Asked Questions (FAQs)