# 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 It might be outdated or ideologically biased.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 log2 (x – 5) + log2 (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.

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Denote the positive extraneous root by [absolute value of [[GAMMA]].sub.ext].
A simple analysis shows that for [[absolute value of [GAMMA]] [less than or equal to] [square root of [1-sin [beta]]/2 [equivalent to] [[absolute value of [GAMMA]].sub.0] the extraneous root [[absolute value of [GAMMA]].sub.ext] will always be greater than or equal to [[absolute value of [GAMMA]], and thus the reflection coefficient [[absolute value of [GAMMA]] will always be given by [[absolute value of [GAMMA]].sub.2] because [[absolute value of [GAMMA]].sub.2] [less than or equal to] [[absolute value of [GAMMA]].sub.1].
The phase error [DELTA][[phi].sub.per] introduced when the extraneous root is taken as the reflection coefficient will be
(4), (5), and (12) with the extraneous root [{[[absolute value of [GAMMA]].sup.2] + 2[[absolute value of [GAMMA]][cos [psi] + sin ([psi] - [beta])] +2 (1 - sin [beta])}.sup.1/2] in place of [[absolute value of [GAMMA]]; 2[pi] is added or subtracted to overcome the 2[pi]-discontinuity problem at the boundary between the first and the forth quadrant.
Note that there is no interest in the extraneous root x = 0.
Extraneous roots are an unsolved mystery for our freshmen level college math students.
Key words: Extraneous roots, Geometry of Extraneous Roots, x Intercepts
Most high school students and college freshmen struggle with the concept of extraneous roots in an algebra course (1).
Unfortunately, extraneous roots are introduced during the algebraic manipulation when an equation is squared to remove the square root present.
Four areas are discussed more thoroughly - multiplicity of roots, extraneous roots, literal equation and trigonometric equation.
Disregarding may result in extraneous roots. Different variants are used in the textbooks to obtain correct final answers.
Table 1: EBDs (Simplification) EBD Problem type CAS = SCH < MATH Forbidden branches are not recorded (CAS, SCH) CAS < SCH = MATH Forbidden branches are recorded (SCH), not recorded (CAS)Absolute value, all branches (SCH) SCH < CAS = MATH [square root of [a.sup.2]] [right arrow] a (SCH) Table 2: EBDs (Equations) EBD Problem type CAS = SCH < MATH Multiplicity of roots 1 (CAS, SCH)Literal equation 1 branch (CAS, SCH) CA:S < SC'H =SWATH Multiplicity of roots 1 (CAS), 2 (SCH)Literal equation 1 branch (CAS), all branches (SCH) Particular solution of trigonometric equation (CAS) SCH < CAS = MATH Multiplicity of roots 1 (SCH), 2 (CAS)Literal equation 1 branch (SCH), all branches (CAS) SCH = MATH < CAS Extraneous roots

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