Solucionador de ecuaciones cúbicas dedicado con raíces reales y complejas, pasos del método Cardano, gráficas cúbicas y ejemplos resueltos.
Calculadora de Cúbica Deprimida
Ingrese los coeficientes de su polinomio arriba y haga clic en "Deprimir cúbica" para ver los resultados.A depressed cubic is a cubic equation that has been simplified by removing the x² term, resulting in the cleaner form t³ + pt + q = 0. This reduction is achieved through the Tschirnhaus substitution x = t − b/(3a), which shifts the polynomial so the second-degree coefficient vanishes. The technique is fundamental to every analytical method for solving cubics.
The depression process transforms four coefficients (a, b, c, d) into just two essential parameters (p, q). Specifically: p = (3ac − b²) / (3a²) and q = (2b³ − 9abc + 27a²d) / (27a³). This simplification is what makes Cardano's formula, Vieta's substitution, and the trigonometric method feasible.
Geometrically, depressing a cubic is equivalent to horizontally translating the graph so that the inflection point sits on the y-axis. This centering eliminates asymmetry in the curve and reveals the polynomial's essential shape, making both algebraic and graphical analysis cleaner.
Every Cardano's method solution begins with depressing the cubic. This calculator automates that critical first step.
Depressing centers the cubic's inflection point at the y-axis, making graphical analysis and symmetry identification easier.
Students learning polynomial transformations use depression to understand how substitutions simplify equation structure.
The substitution is x = t MINUS b/(3a), not plus. Getting the sign wrong shifts the curve in the wrong direction.
The formulas for p and q assume proper normalization. If a ≠ 1, you must include a in the denominators.
p involves 3ac − b² while q involves 2b³ − 9abc + 27a²d. Mixing them up produces completely wrong depressed coefficients.
| Substitution | x = t − b/(3a) |
| Result Form | t³ + pt + q = 0 |
| p formula | (3ac − b²) / (3a²) |
| q formula | (2b³ − 9abc + 27a²d) / (27a³) |
| Geometric Effect | Centers inflection point on y-axis |
Porque se ha "reducido" la complejidad de la ecuación al eliminar el término de grado dos.
Sí, los desplaza horizontalmente. Una vez que encuentres<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">t</span>, debes volver a sumar el factor de desplazamiento para encontrar<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">incógnita</span>.
Sí. Cada ecuación cúbica estándar se puede desplazar para eliminar su<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>término.
La forma estándar es t³ + pt + q = 0, que no tiene término al cuadrado.
La fórmula de Cardano sólo se aplica directamente a la forma deprimida. Al eliminar el término al cuadrado, el álgebra se vuelve lo suficientemente manejable como para derivar una solución en forma cerrada.
