Algebraic solution

An algebraic solution is an algebraic expression which is the solution of an algebraic equation in terms of the coefficients of the variables. It is found only by addition, subtraction, multiplication, division, and the extraction of roots (square roots, cube roots, etc.).

The most well-known example is the solution of the general quadratic equation.

(where a ≠ 0).

There is a more complicated solution for the general cubic equationNickalls, R. W. D., "A new approach to solving the cubic: Cardano's solution revealed," Mathematical Gazette 77, November 1993, 354-359. and quartic equation.Carpenter, William, "On the solution of the real quartic," Mathematics Magazine 39, 1966, 28-30.  The Abel-Ruffini theoremJacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover,  states that the general quintic equation does not have an algebraic solution. This means that the general polynomial equation of degree n, for n ≥ 5, cannot be solved by using algebra. However, under certain conditions, we can get algebraic solutions; for example, the equation  can be solved as 

References

Category:Algebra