This article is about cubic equations in one variable. For cubic equations in two variables, see cubic plane curve.
Not to be confused with Cubic function.
Graph of a cubic function with 3 real roots (where the curve crosses the horizontal axis at y = 0). The case shown has two critical points. Here the function is .
In algebra, a cubic equation in one variable is an equation of the form
in which a is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means:
algebraically: more precisely, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations, square roots and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.)
trigonometrically
numerical approximations of the roots can be found using root-finding algorithms such as Newton's method.
The coefficients do not need to be real numbers. Much of what is covered below is valid for coefficients in any field with characteristic other than 2 and 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are irrational (and even non-real) complex numbers.
In algebra, a cubicequation in one variable is an equation of the form a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} in which a is...
Cubicequations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and...
domain is restricted to the real numbers. Setting f(x) = 0 produces a cubicequation of the form a x 3 + b x 2 + c x + d = 0 , {\displaystyle ax^{3}+bx^{2}+cx+d=0...
quintic and higher-order equations, beyond trivial or special cases. Linear equation Quadratic equationCubicequation Quintic equation Polynomial Newton's...
In mathematics, a cubic plane curve is a plane algebraic curve C defined by a cubicequation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} applied to...
quadratic equations with positive roots, and many cubicequations, although it is not known if they were able to reduce the general cubicequation. Ancient...
mathematician who first discovered a method to solve the depressed cubicequation. Scipione del Ferro was born in Bologna, in northern Italy, to Floriano...
found several solutions of the cubicequation. Omar Khayyam found the general geometric solution of a cubicequation.[citation needed] Omar Khayyam (c...
is most notable for his work on the classification and solution of cubicequations, where he provided a geometric formulation based on the intersection...
possible except for the depressed equation y4 = 0. Now, if m is a root of the cubicequation such that m ≠ 0, equation (1) becomes ( y 2 + p 2 + m ) 2 =...
whose roots are also roots of the original polynomial. The general cubicequation a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} with...
study of equations of state, and was the starting point of cubicequations of state, which most famously continued via the Redlich–Kwong equation of state...
where all vertices have degree 3 Cubic plane curve (mathematics), a plane algebraic curve C defined by a cubicequationCubic reciprocity (mathematics - number...
2 + k 2 x + k 3 {\displaystyle y(x)=k_{1}x^{2}+k_{2}x+k_{3}} for a cubicequation of degree n = 3 {\displaystyle n=3} , f ( x ) = x 3 + a 2 x 2 + a 1...
The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it is not fully empirical but a set of...
Functional equation Functional equation (L-function) Constitutive equation Laws of science Defining equation (physical chemistry) List of equations in classical...
_{r}+c\rho _{r}^{2}+f\rho _{r}^{5}} The three-term virial equation or a cubic virial equation of state Z = 1 + B ρ + C ρ 2 {\displaystyle Z=1+B\rho +C\rho...
the cubicequations by promising not to publish them. Tartaglia divulged the secrets of the solutions of three different forms of the cubicequation in...
of a cubicequation, as he had neither complex numbers at his disposal, nor the algebraic notation to be able to describe a general cubicequation. With...
linear equation for degree one quadratic equation for degree two cubicequation for degree three quartic equation for degree four quintic equation for degree...
theory. Solving quadratic equations with continued fractions Linear equationCubic function Quartic equation Quintic equation Fundamental theorem of algebra...
volumes Conversion of units § Volume Cube (arithmetic) Cube root CubicequationCubic function NIST Guide to the SI - B.8 Factors for Units Listed Alphabetically...
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only...
cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubicequations...
Global Information
