Solucionador de equações cúbicas dedicado com raízes reais e complexas, etapas do método Cardano, gráficos cúbicos e exemplos resolvidos.
Calculadora das Fórmulas de Vieta
Insira seus coeficientes polinomiais acima e clique em "Aplique as fórmulas de Vieta" para ver os resultados.Vieta's formulas establish elegant relationships between the roots of a polynomial and its coefficients, without requiring you to solve the equation first. For a cubic ax³ + bx² + cx + d = 0 with roots r&sub1;, r&sub2;, r&sub3;, the formulas state: the sum of roots r&sub1;+r&sub2;+r&sub3; = −b/a, the sum of pairwise products r&sub1;r&sub2;+r&sub1;r&sub3;+r&sub2;r&sub3; = c/a, and the product of all roots r&sub1;r&sub2;r&sub3; = −d/a.
These formulas are named after François Viète (1540–1603), a French mathematician who pioneered using letters for unknowns. The formulas arise naturally from expanding the factored form a(x−r&sub1;)(x−r&sub2;)(x−r&sub3;) and comparing coefficients with the standard form. They work identically whether the roots are real or complex.
Vieta's formulas serve two critical purposes: error checking (verify that your computed roots are consistent with the original coefficients) and indirect computation (compute symmetric functions of the roots without knowing the roots individually). They are foundational in competition mathematics, abstract algebra, and numerical analysis.
After solving a cubic, check that the sum and product of your roots match −b/a and −d/a respectively.
Many olympiad problems ask about symmetric functions of roots without requiring you to find the roots explicitly.
Vieta's formulas help detect numerical instability — if computed roots don't satisfy the formulas, precision is lost.
The sum of roots is NEGATIVE b/a, and the product is NEGATIVE d/a. Missing these minus signs is extremely common.
All formulas require dividing by the leading coefficient a. If a ≠ 1, the raw coefficient is NOT the answer.
Vieta's formulas work identically for complex roots. The sum and product relationships hold universally.
| Sum of Roots | r&sub1;+r&sub2;+r&sub3; = −b/a |
| Pairwise Products | r&sub1;r&sub2;+r&sub1;r&sub3;+r&sub2;r&sub3; = c/a |
| Product of Roots | r&sub1;·r&sub2;·r&sub3; = −d/a |
| Named After | François Viète (1540–1603) |
| Works With | Both real and complex roots |
Sim! As regras de Vieta aplicam-se perfeitamente mesmo quando as raízes envolvem números imaginários. As partes complexas simplesmente se cancelam durante a adição.
Não, apenas informa como eles se relacionam como um conjunto completo.
Porque as fórmulas de Vieta dependem inerentemente da normalização do polinômio (tornando o coeficiente principal 1) primeiro.
Você pode verificar que a soma das raízes é igual a -b/a, a soma do produto aos pares é igual a c/a e o produto de todas as raízes é igual a -d/a. É uma ferramenta poderosa de verificação de erros.
François Viète foi um matemático francês do século 16 que foi pioneiro no uso de letras para representar incógnitas. Suas fórmulas que conectam raízes a coeficientes continuam sendo a pedra angular da álgebra.
