Solve cubic equations of the form ax³ + bx² + cx + d = 0 using Cardano's formula.
Discriminant (Δ): -0.037037
3 distinct real roots
Coefficients must be valid numbers; 'a' cannot be zero for a cubic equation.
A cubic equation is a polynomial equation of degree three, meaning the highest power of the variable is 3. The general form is ax³ + bx² + cx + d = 0, where a ≠ 0.
Unlike quadratic equations, cubic equations always have at least one real root. They may have three real roots or one real root and two complex conjugate roots, depending on the discriminant.
First, we normalize the equation by dividing by 'a' to get: x³ + px² + qx + r = 0
Q = (3q - p²) / 9
R = (9pq - 27r - 2p³) / 54
Δ = Q³ + R² (discriminant)
1 real, 2 complex roots
All real, repeated roots
3 distinct real roots
Solve: x³ - 6x² + 11x - 6 = 0
Step 1: Calculate Q and R using the formulas
Step 2: Compute discriminant Δ = Q³ + R²
Step 3: Since this factors nicely: (x-1)(x-2)(x-3) = 0
Roots: x₁ = 1, x₂ = 2, x₃ = 3
No! Unlike quadratic equations, every cubic equation with real coefficients has at least one real root. This is guaranteed by the Intermediate Value Theorem.
The discriminant (Δ) determines the nature of the roots. It tells us whether we have three real roots, one real root with two complex conjugates, or repeated roots.
Yes, if the cubic factors nicely! Try the Rational Root Theorem first to find one root, then use polynomial division to reduce to a quadratic.
Gerolamo Cardano published it in 1545, though it was actually discovered by Scipione del Ferro and Niccolò Tartaglia. This was a major breakthrough in Renaissance mathematics.
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