Solve a Cubic Equation
Enter the coefficients a, b, c, and d to solve the cubic equation ax^3 + bx^2 + cx + d = 0. The calculator returns all real and complex roots, along with the algebraic breakdown used to solve the polynomial.
Solving Third-Degree Polynomials
The Cubic Equation Calculator is a highly advanced algebraic tool that solves for all three roots of a third-degree polynomial. While simple cubics can sometimes be factored by grouping, complex cubics require advanced mathematical techniques like Cardano's Formula.
Where:
a, b, c=
The multipliers of the variable termsd=
The fixed numerical valuex=
The values of x that make the equation trueThe Mathematics Behind the Magic
To solve a cubic equation algebraically, the calculator performs a series of transformations:
- Depression: It first converts the standard cubic into a "depressed cubic" of the form $t^3 + pt + q = 0$ by substituting $x = t - \frac{b}{3a}$. This eliminates the squared term.
- Discriminant Calculation: It calculates the discriminant $\Delta$ to determine the nature of the roots.
- Root Extraction: It uses cube roots and trigonometry (for the Casus Irreducibilis scenario where $\Delta < 0$) to find the exact values.
Real-World Uses
- Fluid Dynamics: Solving equations of state for real gases (like the Van der Waals equation).
- Engineering: Finding the deflection curves of beams under continuous loads.
- Economics: Modeling complex cost and profit functions where returns diminish and then accelerate.