strange root

In solving certain of equations, one may obtain besides the proper () roots ( 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.



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


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


are not equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( (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