Enter the coefficients of a cubic equation to calculate one real root using Cardano’s method. This calculator is designed to find one real root for cubic equations of the form ax^3 + bx^2 + cx + d = 0 (with a ≠ 0). If the cubic has three real roots, the calculator will return one of them.
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Cardano’s Formula
When the discriminant \(h \ge 0\), you can compute a real root of a cubic equation using the real-radical form of Cardano’s formula:
x = \sqrt[3]{-\frac{g}{2} + \sqrt{\frac{g^2}{4} + \frac{f^3}{27}}} + \sqrt[3]{-\frac{g}{2} - \sqrt{\frac{g^2}{4} + \frac{f^3}{27}}} - \frac{b}{3a}
Where:
- f = ((3c)/a – (b^2)/(a^2)) / 3 = (3ac – b^2)/(3a^2)
- g = ((2b^3)/(a^3) – (9bc)/(a^2) + (27d)/a) / 27 = (2b^3 – 9abc + 27a^2d)/(27a^3)
- h = g^2/4 + f^3/27
If \(h<0\), the cubic has three real roots; a trigonometric form (rather than real radicals) is typically used to compute real roots. The calculator above will still return one real root in that case.
To calculate one real root of the cubic equation, follow these steps:
How to Calculate a Root Using Cardano’s Formula?
The following steps outline how to calculate one real root of a cubic equation using Cardano’s formula.
- First, determine the coefficients of the cubic equation (a, b, c, d) with a ≠ 0.
- Next, calculate the values of f, g, and h using the coefficients.
- Use h to determine the root case: if h > 0 there is one real root; if h = 0 there are three real roots with at least two equal; if h < 0 there are three distinct real roots.
- Compute one real root: use the real-radical expression when h ≥ 0, and a trigonometric form when h < 0.
- Use the calculator above to verify your result.
Example Problem:
Use the following coefficients as an example problem to test your knowledge.
Coefficient a = 1
Coefficient b = -6
Coefficient c = 11
Coefficient d = -6
