Cubic Equation Solver
Solve cubic equations (ax³ + bx² + cx + d = 0) and find all three roots.
Solve ax³ + bx² + cx + d = 0
What Is the Cubic Equation Solver?
A cubic equation is a polynomial equation of degree 3 in the form ax³ + bx² + cx + d = 0, where a ≠ 0. Every cubic equation with real coefficients has at least one real root and up to three real roots. This solver uses Cardano's formula for one-real-root cases and the trigonometric (Viète) method when all three roots are real.
Formula
How to Use
Enter the four coefficients a, b, c, and d for ax³ + bx² + cx + d = 0. All four fields are required; a must be non-zero. Click Calculate to find all roots. The tool shows the depressed cubic form, the discriminant, and the step-by-step derivation of each root.
Example Calculation
Solve x³ − 6x² + 11x − 6 = 0 (a=1, b=−6, c=11, d=−6). The discriminant Δ > 0, so three real roots exist. Roots: x = 1, x = 2, x = 3 Verification: (x−1)(x−2)(x−3) = x³ − 6x² + 11x − 6 ✓
Understanding Cubic Equation
Cubic equations appear throughout mathematics, physics, and engineering. Finding the volume of a sphere given its surface area, computing the intersection of a line with a cubic Bézier curve, and solving certain kinematics problems all reduce to cubic equations.
The key insight in solving cubics is the substitution x = t − b/(3a), which eliminates the quadratic term and produces the simpler "depressed" form t³ + pt + q = 0. From there, the discriminant Δ = −4p³ − 27q² determines the nature of the roots.
When Δ > 0, all three roots are real and distinct. The trigonometric (Viète) method expresses them as t_k = 2√(−p/3) · cos(⅓ arccos(3q/(2p) · √(−3/p)) − 2πk/3) for k = 0, 1, 2, avoiding complex intermediate values.
When Δ < 0, one root is real and two are complex conjugates, computed via Cardano's formula using the cube root of complex numbers.
Frequently Asked Questions
What is a cubic equation?
A cubic equation is a polynomial of degree 3: ax³ + bx² + cx + d = 0. It always has exactly three roots (counting multiplicity) in the complex numbers.
What is Cardano's formula?
Cardano's formula (1545) gives the roots of a depressed cubic t³ + pt + q = 0 in closed form using cube roots. It is the cubic analogue of the quadratic formula.
When does a cubic have three real roots?
When the discriminant Δ = −4p³ − 27q² > 0 (of the depressed cubic), there are three distinct real roots, found using the trigonometric method.
Can the coefficient a be zero?
No. If a = 0, the equation reduces to a quadratic. Use the Quadratic Equation Solver instead.
How accurate are the results?
Roots are computed using JavaScript's 64-bit floating-point arithmetic, accurate to about 15 significant digits. Very small or very large coefficients may have reduced precision.
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