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Key Features
Instant Calculation
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Unlike the quadratic formula, solving a cubic equation requires checking which of three distinct cases applies — one real root with a complex pair, a repeated real root, or three distinct real roots.
This calculator solves any cubic equation using Cardano's formula, automatically detecting and handling all three cases.
How it works
Enter the coefficients a, b, c, and d. The calculator reduces the equation to a depressed cubic, computes the discriminant, and applies Cardano's formula (or the trigonometric method for three real roots) based on the discriminant's sign.
- Enter a (coefficient of x³).
- Enter b (coefficient of x²).
- Enter c (coefficient of x).
- Enter d (constant).
- Click Calculate to see your results.
Examples
x³-6x²+11x-6=0
This equation has three distinct real roots: 1, 2, and 3 — found using the trigonometric method since the discriminant is negative.
Who should use it
- Solving cubic equations for algebra or numerical methods coursework.
- Checking a hand-solved cubic equation's roots.
Industry applications
- Mathematics and engineering education
- Numerical methods and applied mathematics
Advantages
- Handles all three discriminant cases automatically, including the tricky trigonometric method for three real roots.
- Shows the discriminant value so you can understand which case applies.
Limitations
- Complex roots are reported in a ± bi form rather than as fully separate interactive values.
Common mistakes to avoid
- Assuming every cubic has three real roots — many have just one real root and a complex conjugate pair.
- Manually trying to guess the case instead of trusting the discriminant.
Best practices
- Verify a computed root by substituting it back into the original equation and confirming it evaluates to (approximately) zero.
Tips
- Every cubic equation with real coefficients has at least one real root — the discriminant only determines whether there are one or three real roots total, never zero.