Solucionador de equações cúbicas dedicado com raízes reais e complexas, etapas do método Cardano, gráficos cúbicos e exemplos resolvidos.
Calculadora de Cúbica Deprimida
Insira seus coeficientes polinomiais acima e clique em "Deprimir Cúbico" para ver os 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 a equação foi "reduzida" em complexidade ao eliminar o termo de grau dois.
Sim, isso os desloca horizontalmente. Depois de encontrar<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">t</span>, você deve adicionar novamente o fator de deslocamento para encontrar<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x</span>.
Sim. Toda equação cúbica padrão pode ser deslocada para eliminar seu<span class="font-mono text-primary-700 bg-primary-50 px-1 rounded">x²</span>prazo.
A forma padrão é t³ + pt + q = 0, que não possui termo quadrado.
A fórmula de Cardano só se aplica diretamente à forma deprimida. Ao remover o termo quadrado, a álgebra torna-se suficientemente manejável para derivar uma solução de forma fechada.
