Cubic Equation Calculator – Guide & Formulas
Solve any cubic equation ax³ + bx² + cx + d = 0. Find all real and complex roots with step-by-step Cardano's method solutions.
Solve any cubic equation ax³ + bx² + cx + d = 0 with our free calculator. Enter coefficients to find all three roots including real and complex solutions with step-by-step breakdown.
Key Takeaway
Use the free Cubic Equation Calculator to solve any cubic equation ax³ + bx² + cx + d = 0. find all real and complex roots with step-by-step cardano's method solutions. Get instant results with step-by-step explanations.
How to Use the Cubic Equation Calculator
- Enter coefficient a (cubic term).
- Enter coefficient b (quadratic term).
- Enter coefficient c (linear term).
- Enter coefficient d (constant term).
- Review all three roots with real and imaginary components and step-by-step Cardano's method.
The Formula
Variable Definitions
- a: Coefficient of the cubic term (x³)
- b: Coefficient of the quadratic term (x²)
- c: Coefficient of the linear term (x)
- d: Constant term
- p: Depressed cubic coefficient: (3ac - b²)/(3a²)
- q: Depressed cubic constant: (2b³ - 9abc + 27a²d)/(27a³)
- Δ: Discriminant: -(4p³ + 27q²). Determines root types.
Solving x³ - 6x² + 11x - 6 = 0
Find all roots of the cubic equation with a=1, b=-6, c=11, d=-6.
- Step 1: Substitute x = t + 2 (since b/(3a) = 6/3 = 2). New equation: t³ - t = 0.
- Step 2: p = (3·1·11 - 36)/(3·1) = -3/3 = -1, q = (2(-216) - 9·1·(-6)·11 + 27·(-6))/(27) = 0.
- Step 3: Discriminant Δ = -(4(-1)³ + 27·0²) = 4 > 0, so three distinct real roots.
- Step 4: t³ - t = 0 → t(t² - 1) = 0 → t = 0, t = 1, t = -1.
- Step 5: Back-substitute: x = t + 2 → x = 2, x = 3, x = 1.
Frequently Asked Questions
What is a cubic equation?
A cubic equation is a polynomial equation of degree 3: ax³ + bx² + cx + d = 0. It has exactly three roots (counting multiplicity and complex roots).
How many real roots can a cubic have?
A cubic equation always has at least one real root. It can have either one real root and two complex conjugate roots, or three real roots (which may include repeated roots).
What is Cardano's formula?
Cardano's formula is an explicit formula for solving cubic equations. It expresses the roots in terms of the coefficients using cube roots and square roots, similar to the quadratic formula for degree 2.
What is the discriminant of a cubic?
The discriminant Δ = 18abcd - 4b³d + b²c² - 4ac³ - 27a²d² determines root nature: Δ > 0 means three distinct real roots, Δ = 0 means repeated roots, Δ < 0 means one real and two complex roots.
What is a depressed cubic?
A depressed cubic is t³ + pt + q = 0, obtained by substituting x = t - b/(3a). This eliminates the quadratic term and simplifies Cardano's method.
Can I solve cubics by factoring?
Yes, if the cubic factors nicely. For example, x³ - 6x² + 11x - 6 = (x-1)(x-2)(x-3). But most cubics do not factor neatly and require Cardano's formula.
What is Vieta's formula for cubics?
For roots r₁, r₂, r₃: r₁ + r₂ + r₃ = -b/a, r₁r₂ + r₁r₃ + r₂r₃ = c/a, r₁r₂r₃ = -d/a.
How does this differ from a quadratic?
Quadratic equations (degree 2) have at most 2 roots and use the quadratic formula. Cubic equations (degree 3) have exactly 3 roots and require Cardano's method or numerical approaches.