# Extraneous Root

## extraneous root

[ik¦strān·ē·əs ′rüt] (mathematics)

A root that is introduced into an equation in the process of solving another equation, but is not a solution of the equation to be solved.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

The following article is from

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.## Extraneous Root

a root, or solution, of an intermediate equation—an equation obtained in the process of solving a given equation—that is not a root of the given equation. Extraneous roots appear because in solving an equation we cannot always pass to equivalent equations when we simplify it. They may arise, for example, in raising both sides of an equation to a power, in clearing an equation of fractions, or in taking antilogarithms. Thus, the equation log_{2} (*x* – 5) + log_{2} (*x* – 3) = 3 has the single root *x* = 7. If, however, we take antilogarithms, we obtain the equation (*x* – 5)(*x* – 3) = 8. It has not only the root *x* = 7 but also the root *x*= 1, which is an extraneous root of the initial equation.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.