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