strange root


In solving certain of equations, one may obtain besides the proper () roots (http://planetmath.org/Equation) also some strange roots which do not satisfy the original equation. Such a thing can happen especially when one has in some stage squared both sides of the treated equation; in this situation one must check all “roots” by substituting them to the original equation.

Example.

x-x=12
x-12=x
(x-12)2=(x)2
x2-24x+144=x
x2-25x+144=0
x=25±252-41442=25±72
x=16x=9

Substituting these values of x into the left side of the original equation yields

16-4=12,9-3=6.

Thus, only  x=16  is valid,  x=9  is a strange root. (How  x=9  is related to the solved equation, is explained by that it may be written  (x)2-x-12=0, from which one would obtain via the quadratic formula that  x=1±72,  i.e.  x=4  or  x=-3.  The latter corresponds the value  x=9,  but it were relevant to the original equation only if we would allow negative values for square roots of positive numbers; the practice excludes them.)

The general explanation of strange roots when squaring an equation is, that the two equations

a=b,
a2=b2

are not equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/Equivalent3) (but the equations  a=±b  and  a2=b2  would be such ones).

Title strange root
Canonical name StrangeRoot
Date of creation 2013-03-22 17:55:53
Last modified on 2013-03-22 17:55:53
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 97D99
Classification msc 26A09
Synonym wrong root
Synonym extraneous root
Related topic QuadraticFormula
Related topic LogicalOr
Related topic SquaringConditionForSquareRootInequality