Cubic Equation Solver
Find solutions to third-degree polynomial equations ax³ + bx² + cx + d = 0. Implements Cardano’s trigonometric and algebraic formulas to resolve real and complex root configurations.
Leading cubic coefficient (non-zero)
Quadratic coefficient
Linear coefficient
Constant term
Primary real root guaranteed by Intermediate Value Theorem
How to Use This Calculator
- Enter the leading coefficient a (for x³).
- Enter coefficients b, c, and constant d.
- View the real root solutions and discriminant immediately.
Formula & Calculation Methodology
Cardano’s method depresses the cubic by eliminating the quadratic term into t³ + pt + q = 0, followed by algebraic or trigonometric radical extraction.
Worked Example
Solving x³ − 6x² + 11x − 6 = 0
Solving x³ − 1 = 0
Frequently Asked Questions
Does every cubic equation have at least one real root?↓
Yes. Because odd-degree polynomials extend from −∞ to +∞, the graph must cross the x-axis at least once by the Intermediate Value Theorem.
What is the "casus irreducibilis"?↓
When a cubic has three distinct real roots, Cardano’s algebraic formula involves square roots of negative numbers, necessitating trigonometric functions to extract real answers.
What if coefficient a is 0?↓
If a = 0, the equation reduces to a quadratic equation bx² + cx + d = 0, which can be solved via the quadratic formula.
What is a limitation of floating cubic calculations?↓
Near multiple roots where the discriminant is close to zero, numerical precision can fluctuate slightly due to floating point roundoff.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.