Why Multiple Roots?

Date: 9/9/96 at 9:47:27
From: Anonymous
Subject: Why Multiple Roots?

I have been tutoring a high school algebra student who asked me 
the following question, "Why do I have to check the answer of a 
radical equation?"  I hadn't really thought about it before and didn't 
know what to tell her. 
 
I looked in a couple algebra books and they said that raising an 
equation to an even power sometimes gives an extraneous root.  That 
still doesn't explain WHY.

I think the reason our answer sometimes doesn't satisfy the 
original equation is because we only consider the principle nth root 
(positive) and don't consider the negative square roots.  Is that the 
right explanation?  If not, please explain.

Date: 9/9/96 at 11:33:57
From: Doctor Robert
Subject: Re: Why Multiple Roots?

When you multiply both sides of an equation by an expression which 
includes the variable, you run the risk of introducing extraneous 
roots.  Let me give a very simple example.  Suppose you have the 
equation

   x = 3

for which the solution is obviously 3.  Now multiply both sides of 
this equation by x which gives 

   x^2 = 3x

One of the roots of this equation is still 3, but there is also 
another root, namely zero, which has been introduced.  {0,3} are roots 
of the second equation, but only 3 is a root of the first. Often, in 
the process of solving radical equations, we square both sides -
the equivalent of multiplying by an expression which includes the 
variable. We find roots to this new equation, but sometimes not all 
roots of the new equation are also roots of the original equation. 
I hope that this helps.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/