Cubic Equation Calculator
Solve cubic equations using Cardano's formula with step-by-step solutions
Solve Cubic Equation
Current Equation:
x³ = 0
General form: ax³ + bx² + cx + d = 0
x³ coefficient
x² coefficient
x coefficient
constant term
Solution Results
Discriminant
Δ = 0.00000
Triple root (all three roots are equal)
Roots
Factorization
1(x - 0.00000)³
Method Used
Direct formula for triple root
Example Calculations
Example 1: x³ - 6x² + 11x - 6 = 0
Coefficients: a = 1, b = -6, c = 11, d = -6
Discriminant: Δ = 0 (three real roots with at least two equal)
Roots: x₁ = 1, x₂ = 2, x₃ = 3
Factorization: (x - 1)(x - 2)(x - 3)
Example 2: x³ - 2x² - 3x + 2 = 0
Coefficients: a = 1, b = -2, c = -3, d = 2
Method: Use Cardano's formula
Result: One real root and two complex conjugate roots
Note: This requires advanced calculation methods
Example 3: x³ = 0
Coefficients: a = 1, b = 0, c = 0, d = 0
Type: Triple root
Root: x₁ = x₂ = x₃ = 0
Factorization: x³
Root Types by Discriminant
Δ > 0
Three distinct real roots
Δ < 0
One real, two complex conjugate roots
Δ = 0
Three real roots, at least two equal
Cardano's Formula
Discriminant:
Δ = b²c² - 4ac³ - 4b³d - 27a²d² + 18abcd
Q and R values:
Q = (3ac - b²)/(9a²)
R = (9abc - 27a²d - 2b³)/(54a³)
Root formula:
x = S + T - b/(3a)
Where S and T are complex cube roots
Solving Tips
Try rational root test first for integer coefficients
Look for obvious roots like ±1, ±2 by substitution
Factor out common terms when possible
Use synthetic division once you find one root
Cardano's formula always works but can be complex
Understanding Cubic Equations
What is a Cubic Equation?
A cubic equation is a polynomial equation of degree 3, having the general form ax³ + bx² + cx + d = 0, where a ≠ 0. Every cubic equation has exactly three complex roots (counting multiplicity), with at least one real root.
Methods for Solving
- •Rational Root Test: Test possible rational roots
- •Factoring: Look for common factors or patterns
- •Cardano's Formula: Universal method using cube roots
- •Synthetic Division: Reduce to quadratic after finding one root
Cardano's Formula
For ax³ + bx² + cx + d = 0:
Q = (3ac - b²)/(9a²)
R = (9abc - 27a²d - 2b³)/(54a³)
S = ∛(R + √(Q³ + R²))
T = ∛(R - √(Q³ + R²))
x₁ = S + T - b/(3a)
Applications
- •Volume and surface area problems
- •Physics: motion and acceleration equations
- •Economics: cost and profit optimization
- •Engineering: stress and strain analysis
The Discriminant
The discriminant Δ = b²c² - 4ac³ - 4b³d - 27a²d² + 18abcd tells us about the nature of the roots before we calculate them:
Δ > 0
Three distinct real roots
Δ < 0
One real root, two complex conjugates
Δ = 0
Multiple roots (at least two equal)