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This is a cubic equation in y. Solve for y using any method for solving such equations (e.g. conversion to a reduced cubic and application of Cardano's formula). Any of the three possible roots will do.
be the general quartic equation we want to solve. Dividing by a 4, provides the equivalent equation x 4 + bx 3 + cx 2 + dx + e = 0, with b = a 3 / a 4 , c = a 2 / a 4 , d = a 1 / a 4 , and e = a 0 / a 4 . Substituting y − b / 4 for x gives, after regrouping the terms, the equation y 4 + py 2 + qy + r = 0, where
An example of using Newton–Raphson method to solve numerically the equation f(x) = 0. In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign.
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
[2] [3] The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. [4] Solving an equation containing variables ...
John Renze, Eric W. Weisstein, "Separation of variables" ("Differential Equation") at MathWorld. Methods of Generalized and Functional Separation of Variables at EqWorld: The World of Mathematical Equations; Examples of separating variables to solve PDEs "A Short Justification of Separation of Variables"
Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.. Solving linear, quadratic, cubic and quartic equations in terms of radicals and elementary arithmetic operations on the coefficients can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulas that yield the required solutions.
The Barth surface, shown in the figure is the geometric representation of the solutions of a polynomial system reduced to a single equation of degree 6 in 3 variables. Some of its numerous singular points are visible on the image. They are the solutions of a system of 4 equations of degree 5 in 3 variables.