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Cubic Equation Calculator

Solve ax^3 + bx^2 + cx + d = 0, see all three roots, and understand whether the cubic has three real roots, a repeated root, or a complex-conjugate pair.

equation
Examples

Three distinct real roots: 1, 2, and 3.

Root pattern
Three distinct real roots
Root x1
1
Root x2
2
Root x3
3
Discriminant Δ
4

Three real roots were found. The solver uses the discriminant branch and checks the reported values by substitution before showing them.

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Examples

How It Works

Formula

ax3+bx2+cx+d=0,a≠0ax^3 + bx^2 + cx + d = 0,\quad a \ne 0

x=t−b3ax = t - \frac{b}{3a}

Δ=18abcd−4b3d+b2c2−4ac3−27a2d2\Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2

Variables

aa

Coefficient of x^3 (must be non-zero)

bb

Coefficient of x^2

cc

Coefficient of x

dd

Constant term

xx

Unknown root value

Start with the canonical cubic ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0 with a≠0a \ne 0. The solver normalizes by dividing through by aa, then shifts the variable so the x2x^2 term disappears. From there the discriminant tells which branch to use: three real roots, a repeated real root, or one real root with a complex-conjugate pair.

This is a focused numeric cubic solver, not a full symbolic algebra system. When a root is not simple enough to present cleanly, the calculator labels it with ≈\approx and checks the reported roots by substitution before showing them.

Frequently Asked Questions

01What does the cubic discriminant tell me?
The discriminant classifies the roots. A positive discriminant means three distinct real roots, zero means at least one repeated real root, and a negative discriminant means one real root plus a complex-conjugate pair.
02Why does the calculator show approximate roots sometimes?
Many cubic roots are not simple fractions or integers. In those cases the calculator shows a numerical approximation and verifies the reported roots by substitution instead of pretending every answer has a neat symbolic form.
03What happens when a = 0?
Then the equation is no longer cubic. This calculator stops and tells you directly instead of leaving an old cubic answer on screen.
04How are repeated roots handled?
Near-equal roots are grouped when the discriminant is effectively zero, so floating-point noise does not invent fake distinctions between values that should repeat.

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