Cubic Equation Solver

How it is calculated

For ax³ + bx² + cx + d = 0 the tool shifts x = t + offset with offset = −b / 3a, giving t³ + pt + q = 0, where p = (3ac − b²) / 3a² and q = (2b³ − 9abc + 27a²d) / 27a³. It then checks D = q²/4 + p³/27. If D is positive, it uses Cardano's cube roots and prints one real root plus a complex pair. If D is zero (within 1e−12), it prints the repeated real roots, or a single value for a triple root. If D is negative, it uses a trigonometric formula for three distinct real roots. All numbers are rounded to two decimals, and the general discriminant Δ is also shown. The coefficient a cannot be 0, since the equation is then not cubic.

Common mistakes

• Leaving a coefficient blank or entering a = 0 for an equation that is really quadratic; the tool gives no result when a is zero.
• Dropping the minus sign of a coefficient, for example entering 6 instead of −6 for the x² term.
• Reading a rounded root such as 1.00 as exact; the roots are shown to two decimals only, so substitute them back before relying on them.
• Expecting every repeated root to be listed three times: a triple root is printed once, and the sign of Δ tells how many roots are real (positive: three distinct real; negative: one real and two complex).