What 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.

Why 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.

What 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.

How 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.

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.

How It Works

Formula

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

x = t - \frac{b}{3a}

\Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2

Start with the canonical cubic $ax^3 + bx^2 + cx + d = 0$ with $a \ne 0$ . The solver normalizes by dividing through by $a$ , then shifts the variable so the $x^2$ 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.

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