basic

Cardano's One Real Root of a Depressed Cubic Calculator

x = ∛(−q/2 + √(q²/4 + p³/27)) + ∛(−q/2 − √(q²/4 + p³/27)).

Find one real root of a depressed cubic t³ + p·t + q = 0 using Cardano's formula in the discriminant-positive case (Δ = q²/4 + p³/27 > 0): t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ). Reports the discriminant, both cube-root terms, and the real root. (Discriminant ≤ 0 means three real roots — use the trigonometric form instead.)

Daniel Oh By Daniel Oh
Published Last reviewed 1 min read

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How to use this calculator

  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ); Δ = q²/4 + p³/27.

In depth

Find one real root of a depressed cubic t³ + p·t + q = 0 using Cardano's formula in the discriminant-positive case (Δ = q²/4 + p³/27 > 0): t = ∛(−q/2 + √Δ) + ∛(−q/2 − √Δ). Reports the discriminant, both cube-root terms, and the real root. (Discriminant ≤ 0 means three real roots — use the trigonometric form instead.)