Cubic Equation Solver – Roots of ax³+bx²+cx+d

Cubics always have at least one real root, and this solver finds all three. For x³ − 6x² + 11x − 6 = 0, rational-root testing tries divisors of 6: x = 1 works (1−6+11−6 = 0), and synthetic division leaves x² − 5x + 6 = (x−2)(x−3), so the roots are 1, 2, 3. Vieta’s relations confirm instantly: sum 6 = −b/a ✓, pairwise sum 11 = c/a ✓, product 6 = −d/a ✓. For non-factorable cases the tool applies the Cardano-style path — depress the cubic via x = t − b/(3a), then solve the resulting t³ + pt + q = 0 — reporting the discriminant Δ = 18abcd − 4b³d + b²c² − 4ac³ − 27a²d²: Δ > 0 means three distinct real roots, Δ = 0 a repeated root, Δ < 0 one real plus a complex-conjugate pair. Each transformation is printed so the method mirrors hand algebra.

1. 1

   Enter a, b, c, d

   Coefficients of ax³ + bx² + cx + d = 0

2. 2

   Test rational roots

   Divisors of d/a are tried with substitution shown

3. 3

   Deflate and solve

   Synthetic division leaves a quadratic, solved exactly

4. 4

   Verify with Vieta

   Sum, pairwise, product checks confirm the root set

• 1

  Every cubic has at least one real root

• 2

  Rational candidates: ± divisors of the constant over the leading coefficient

• 3

  Vieta checks catch transcription errors in seconds

• Dropping a root after the quadratic step — cubics have three (with multiplicity)

• Sign errors in synthetic division

• Applying the quadratic formula to the undepressed cubic

• Forgetting complex roots come in conjugate pairs