statement of the equality of two expressions formulated by applying to a set of variables the algebraic operations, namely, addition, subtraction, multiplication, division, raising to a power, and extraction of a root. Examples are x³+1 and (y⁴x²+2xy–y)/(x–1)=12. An important special case of such equations is that of polynomial equations, expressions of the form axⁿ + bxⁿ⁻¹+…+gx+h=k. They have as many solutions as their degree (n), and the search for their solutions stimulated much of the development of classical and modern algebra. Equations like xsin(x)=c that involve nonalgebraic operations, such as logarithms or trigonometric functions, are said to be transcendental.

The solution of an algebraic equation is the process of finding a number or set of numbers that, if substituted for the variables in the equation, reduce it to an identity. Such a number is called a root of the equation. See also Diophantine equation; linear equation; quadratic equation.