The Cardano’s Formula Calculator helps users find the roots of cubic equations, which are equations of the form:
ax³ + bx² + cx + d = 0
This calculator is particularly useful in fields such as mathematics, engineering, and physics, allowing users to solve complex cubic equations quickly and efficiently without manual calculations.
Formula of Cardano’s Formula Calculator
Cardano’s formula is used to find the roots of a cubic equation by reducing it to a depressed cubic form. Here’s the process:
- Start with the cubic equation:ax³ + bx² + cx + d = 0
- Use the substitution:x = y – (b / 3a)This transforms the equation into:y³ + py + q = 0where:p = (3ac – b²) / (3a²)q = (2b³ – 9abc + 27a²d) / (27a³)
- The roots of the depressed cubic equation y³ + py + q = 0 can be calculated using the formulas:y₁ = C + Δ₀ / Cy₂ = ωC + Δ₀ / (ωC)y₃ = ω²C + Δ₀ / (ω²C)where:C = ³√(Δ₁ / 2 + √(Δ₁² / 4 – Δ₀³ / 27))Δ₀ = -p / 3Δ₁ = -q / 2ω = (-1 + √(-3)) / 2 (a non-real cube root of unity)
- Finally, substitute back to find the roots of the original cubic equation:x = y + (b / 3a)
Common Terms for Cubic Equations
Here’s a table of common terms related to cubic equations that may help users understand the calculations better:
| Term | Description |
|---|---|
| Coefficient | A number multiplying a variable (e.g., a, b, c, d) |
| Root | A solution to the equation (the value of x) |
| Depressed | A simplified form of a cubic equation |
| Unity | A complex number used in calculations |
Example of Cardano’s Formula Calculator
Let’s consider the cubic equation:
2x³ – 4x² + 3x – 6 = 0
Here, the coefficients are:
- a = 2
- b = -4
- c = 3
- d = -6
Using Cardano’s formula, we first calculate:
- p = (3 * 2 * 3 – (-4)²) / (3 * 2²) = (18 – 16) / 12 = 2/12 = 1/6
- q = (2 * (-4)³ – 9 * 2 * (-4) * (-6) + 27 * 2² * (-6)) / (27 * 2³)Calculating q yields q = -146 / 216 = -73 / 108.
Now we proceed to calculate C and the roots y₁, y₂, and y₃ using the given formulas.
Most Common FAQs
Cardano’s formula is a method for finding the roots of cubic equations. It simplifies the process of solving complex polynomial equations.
To use the calculator, enter the coefficients a, b, c, and d of your cubic equation, and click calculate. The calculator will provide you with the roots of the equation.
Yes, the Cardano’s Formula Calculator can handle complex roots, providing solutions even when the roots are not real numbers.