Extraneous Roots

Date: 06/13/2001 at 00:48:26
From: Kostya Tomashevsky
Subject: An equation with a square root

Recently I came across this equation:

  sqrt(x-3) = x-5

I squared both sides to solve the quadratic equation, and got the 
solutions 4 and 7. When I checked them I saw that 4 only worked when 
I calculated the square root as a negative, and 7 only worked when the 
root was assumed positive. My question is, are both of these solutions 
valid, or only 7? Or neither of them?

Thank you,
Kostya Tomashevsky

Date: 06/13/2001 at 08:41:23
From: Doctor Peterson
Subject: Re: An equation with a square root

Hi, Kostya.

What you have discovered is called "extraneous roots." You can read 
about them here:

   Why Multiple Roots?
   http://mathforum.org/dr.math/problems/duff.9.9.96.html   

What happens is that, by squaring both sides of the equation, you have 
made an equation whose roots include those of both

    sqrt(x-3) = x-5

and

    -sqrt(x-3) = x-5

because these two equations give the same result when you square them. 
If you treated the radical in the original problem as having two 
values, positive and negative, then both of the roots you found would 
be valid; but since we interpret a radical as a function that returns 
only the positive root, only one works. You can't be sure which root 
will be valid for the original equation until you check them.

You always have to check each solution you find, and in this situation 
only those that work are valid solutions to the problem.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/