Algebraic solution
Encyclopedia
An algebraic solution is a closed form expression that is the solution of an algebraic equation in terms of the coefficients, relying only on addition
, subtraction
, multiplication
, division
, and the extraction of roots (square roots, cube roots, etc.).
The most well-known example is the solution
introduced in secondary school, of the quadratic equation
(where a ≠ 0).
There exist more complicated algebraic solutions for the general cubic equation and quartic equation. The Abel-Ruffini theorem states that the general quintic equation
lacks an algebraic solution, and this directly implies that the general polynomial equation of degree n, for n ≥ 5, cannot be solved algebraically. However, under certain conditions algebraic solutions can be obtained; for example, the equation
can be solved as 
Algebraic solutions form a subset of closed-form expression
s, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
, multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
, and the extraction of roots (square roots, cube roots, etc.).
The most well-known example is the solution
introduced in secondary school, of the quadratic equation
Quadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
(where a ≠ 0).
There exist more complicated algebraic solutions for the general cubic equation and quartic equation. The Abel-Ruffini theorem states that the general quintic equation
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
lacks an algebraic solution, and this directly implies that the general polynomial equation of degree n, for n ≥ 5, cannot be solved algebraically. However, under certain conditions algebraic solutions can be obtained; for example, the equation
can be solved as Algebraic solutions form a subset of closed-form expression
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
s, because the latter permit transcendental functions (non-algebraic functions) such as the exponential function, the logarithmic function, and the trigonometric functions and their inverses.