Cubic Equation Solver
This cubic equation solver finds all roots once you enter the coefficients a, b, c and d. For x³ − 6x² + 11x − 6 = 0 it returns the three roots x₁ = 1, x₂ = 2 and x₃ = 3.
Example: a (of x³) 1 · b (of x²) -6 · c (of x) 11 · d (constant) -6 → Roots: x₁ = 1 · x₂ = 2 · x₃ = 3. Source: Wikipedia – Cardanische Formeln. Updated: .
How it is calculated
Every cubic equation ax³ + bx² + cx + d = 0 has three roots if complex numbers are allowed: either three real ones or one real and two complex conjugates. The solver finds all three, shows them as a number or in the form a + bi, and checks each by substitution.
How it is calculated
First the equation is divided by a and the substitution x = y − b/(3a) removes the quadratic term, leaving the depressed cubic y³ + p·y + q = 0. Its discriminant Δ = −4p³ − 27q² decides the case: for Δ > 0 there are three distinct real roots, found trigonometrically; for Δ < 0 one real and two complex roots from Cardano's formula; for Δ = 0 there are repeated roots. Real values are then polished with Newton's method.
Linear factors and check
With the roots the polynomial can be written as a(x − x₁)(x − x₂)(x − x₃). With complex roots a real quadratic factor remains. The “Check” column shows the magnitude of f(x) at the root, which is around 10⁻¹⁴ or smaller.
- Example with complex roots: x³ + x + 2 = 0 has x₁ = −1 and x₂,₃ = 0.5 ± 1.3229i.
- With a = 0 the equation would only be quadratic; the solver points that out.
- Rational roots are often found faster by testing the divisors of d and using polynomial division.
Frequently asked questions
How do you solve a third-degree equation?
Either guess one root and reduce to a quadratic with polynomial division, or use Cardano's formulas. The solver takes the second route and returns all three roots.
What does the discriminant tell you for a cubic?
Δ > 0: three distinct real roots. Δ = 0: a repeated root. Δ < 0: one real and two complex roots.
How many roots does x³ − 6x² + 11x − 6 = 0 have?
Three real ones: 1, 2 and 3. The equation can be written as (x − 1)(x − 2)(x − 3) = 0.
Sources and legal basis
- Wikipedia – Cardanische Formeln
- Wikipedia – Cubic equation
- Wikipedia – Diskriminante (Polynom dritten Grades)
As of: · Editorial policy
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