What is a cubic equation?

A cubic equation has the form ax^3 + bx^2 + cx + d = 0, where a, b, c, d are numbers and a 0. Every cubic has exactly three roots (counted with multiplicity): either three real roots, or one real root plus a pair of complex conjugate roots.

How does this calculator solve a cubic?

Two ways, depending on d. If d = 0, then x = 0 is automatically a root — factor out x, leaving a quadratic to solve. If d 0, the calculator searches for a rational root using the Rational Root Theorem, then deflates the cubic by synthetic division into a quadratic factor, which is solved with the quadratic formula.

What is the Rational Root Theorem?

For ax^3 + bx^2 + cx + d = 0 with integer-like coefficients, any rational root must equal p/q, where p divides the constant term d and q divides the leading coefficient a. Testing every such candidate finds a root quickly without guessing.

Why do complex roots show up?

Once a real root is found and factored out, the remaining quadratic factor can have a negative discriminant — which means its two roots are complex conjugates. Every cubic still has at least one real root (a cubic curve always crosses the x-axis at least once), but the other two can be complex.

What are Vieta's formulas, and what does the check verify?

For ax^3 + bx^2 + cx + d = 0 with roots x_1, x_2, x_3: the sum x_1+x_2+x_3 = -b/a, and the product x_1 x_2 x_3 = -d/a (there is also a sum-of-pairs identity, x_1x_2+x_1x_3+x_2x_3 = c/a). This calculator recomputes -b/a and -d/a straight from the coefficients as a sanity check against the three roots it found — real and complex together.

What happens when a = 0?

If a = 0 the x^3 term disappears and the equation is at most quadratic, not cubic. This calculator reports that case as "not cubic"; enter a nonzero a.

Cubic Equation Calculator

Solve any cubic equation ax³ + bx² + cx + d = 0. Get all three roots — real and complex — with steps, plus a Vieta's formulas check.

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The cubic equation

A cubic equation has the form $ax^3 + bx^2 + cx + d = 0$ , where $a$ , $b$ , $c$ , $d$ are numbers and $a \neq 0$ . Unlike a quadratic, there is no single simple formula most people memorize — instead, this calculator finds a root first, then reduces the problem to a quadratic. Every cubic has exactly three roots (counting multiplicity): three real numbers, or one real number and a complex conjugate pair.

Tip: A cubic always has at least one real root — its graph is a curve that runs from

-\infty

+\infty

(or the reverse), so it must cross the x-axis somewhere. The other two roots can be real or complex.

Two solving strategies

This calculator picks its strategy from the constant term $d$ :

Case	Strategy
$d = 0$	Factor out $x$ . Every term has an $x$ , so $x = 0$ is a root immediately: $x(ax^2 + bx + c) = 0$ . What remains is a quadratic.
$d \neq 0$	Rational Root Theorem + deflation. Test candidate roots $\pm p/q$ (divisors of $d$ over divisors of $a$ ) until one works, then divide it out by synthetic division to leave a quadratic factor.

Either way, the last step is solving a quadratic — the same one behind the Quadratic Formula calculator.

Both strategies rely on finding one rational root first, so this calculator handles cubics that have at least one rational root; a cubic that is irreducible over the rationals (no integer or simple-fraction root) needs Cardano's formula or a numerical method, which is beyond this tool's scope.

Worked example — three rational roots

Solve $x^3 - 6x^2 + 11x - 6 = 0$ , so $a=1$ , $b=-6$ , $c=11$ , $d=-6$ .

Test rational roots: since $d = -6 \neq 0$ , try divisors of $6$ over divisors of $1$ : $x = 1$ gives $1 - 6 + 11 - 6 = 0$ . Found.
Deflate: dividing $x^3 - 6x^2 + 11x - 6$ by $(x-1)$ leaves $x^2 - 5x + 6$ .
Solve the quadratic: $x^2 - 5x + 6 = (x-2)(x-3)$ , so the discriminant is $\Delta = 25-24=1 > 0$ , giving roots $x=3$ and $x=2$ .
Three roots: $x_1 = 1$ , $x_2 = 3$ , $x_3 = 2$ — the factorization is $(x-1)(x-2)(x-3)$ .
Vieta check: $1+2+3=6=-b/a$ , and $1 \times 2 \times 3 = 6 = -d/a$ . Both match.

Worked example — d = 0 (factor out x)

Solve $x^3 - 4x^2 + 3x = 0$ , so $a=1$ , $b=-4$ , $c=3$ , $d=0$ .

Factor out x: since $d=0$ , $x = 0$ is a root: $x(x^2 - 4x + 3) = 0$ .
Solve the quadratic: $x^2 - 4x + 3 = (x-1)(x-3)$ , discriminant $\Delta = 16-12=4>0$ , roots $x=1$ and $x=3$ .
Three roots: $x_1 = 0$ , $x_2 = 3$ , $x_3 = 1$ .
Vieta check: $0+3+1=4=-b/a$ , and $0 \times 3 \times 1 = 0 = -d/a$ . Both match — the product is exactly $0$ because $x=0$ is one of the roots.

Worked example — a real root and a complex pair

Solve $x^3 - 8 = 0$ , so $a=1$ , $b=0$ , $c=0$ , $d=-8$ .

Test rational roots: try divisors of $8$ : $x=2$ gives $8 - 8 = 0$ . Found.
Deflate: dividing $x^3 - 8$ by $(x-2)$ leaves $x^2 + 2x + 4$ .
Solve the quadratic: discriminant $\Delta = 4 - 16 = -12 < 0$ , so the remaining two roots are complex conjugates: $x = -1 \pm \sqrt{3}\,i$ .
Three roots: $x_1 = 2$ (real), $x_2 = -1+\sqrt{3}i$ , $x_3=-1-\sqrt{3}i$ (a complex conjugate pair).
Vieta check: $2 + (-1+\sqrt3 i) + (-1-\sqrt3 i) = 0 = -b/a$ (the imaginary parts cancel), and $2 \times \big[(-1)^2+(\sqrt3)^2\big] = 2 \times 4 = 8 = -d/a$ . Both match, even with complex roots in the mix.

Vieta's formulas as a built-in check

For any cubic $ax^3+bx^2+cx+d=0$ with roots $x_1, x_2, x_3$ , Vieta's formulas relate the roots directly to the coefficients — no solving required:

$x_1+x_2+x_3=-\frac{b}{a} \qquad x_1x_2x_3=-\frac{d}{a}$

This calculator shows both identities computed straight from $a$ , $b$ , $d$ , alongside the three roots it found. Because the identities hold for real and complex roots alike, they double as a correctness check: if the sum and product of the roots you calculated by hand don't match $-b/a$ and $-d/a$ , a root was mis-copied somewhere.

Where cubic equations show up

Cubic equations model volume problems (a box's dimensions related by one variable), cubic growth and decay curves, and the inflection points of higher-degree curves. Once you have one root of a cubic — by inspection, a graph, or the Rational Root Theorem — the quadratic formula or completing the square finishes the job on what's left.

Frequently asked questions

What is a cubic equation?: A cubic equation has the form $ax^3 + bx^2 + cx + d = 0$ , where $a$ , $b$ , $c$ , $d$ are numbers and $a \neq 0$ . Every cubic has exactly three roots (counted with multiplicity): either three real roots, or one real root plus a pair of complex conjugate roots.
How does this calculator solve a cubic?: Two ways, depending on $d$ . If $d = 0$ , then $x = 0$ is automatically a root — factor out $x$ , leaving a quadratic to solve. If $d \neq 0$ , the calculator searches for a rational root using the Rational Root Theorem, then deflates the cubic by synthetic division into a quadratic factor, which is solved with the quadratic formula.
What is the Rational Root Theorem?: For $ax^3 + bx^2 + cx + d = 0$ with integer-like coefficients, any rational root must equal $\pm p/q$ , where $p$ divides the constant term $d$ and $q$ divides the leading coefficient $a$ . Testing every such candidate finds a root quickly without guessing.
Why do complex roots show up?: Once a real root is found and factored out, the remaining quadratic factor can have a negative discriminant — which means its two roots are complex conjugates. Every cubic still has at least one real root (a cubic curve always crosses the x-axis at least once), but the other two can be complex.
What are Vieta's formulas, and what does the check verify?: For $ax^3 + bx^2 + cx + d = 0$ with roots $x_1, x_2, x_3$ : the sum $x_1+x_2+x_3 = -b/a$ , and the product $x_1 x_2 x_3 = -d/a$ (there is also a sum-of-pairs identity, $x_1x_2+x_1x_3+x_2x_3 = c/a$ ). This calculator recomputes $-b/a$ and $-d/a$ straight from the coefficients as a sanity check against the three roots it found — real and complex together.
What happens when a = 0?: If $a = 0$ the $x^3$ term disappears and the equation is at most quadratic, not cubic. This calculator reports that case as "not cubic"; enter a nonzero $a$ .