Question 79742: What is an extraneous solutions?
Found 2 solutions by tutor_paul, Edwin McCravy:
Answer by tutor_paul(519)   (Show Source):
You can put this solution on YOUR website! The best way to understand what extraneous roots are is to go through a problem where you get one. The following is a solution to a problem that I did previously which dealt with extraneous roots:
First, get the radical expression alone on one side of the equation:
Now, square both sides of the equation to get rid of the radical:
Simplify the right hand side:
Combine like terms and equate to zero:
Factor the expression:
Equate each factor to zero, and solve:
 and 
Now, this part is important...you need to plug these answers back in to
the original equation to be sure they are not "extraneous." An extraneous
root may be mathematically correct, but it is not the true answer. If you
plug y=1 back into the original equation, you will see that the equation
DOES NOT hold true. Hence, this is an extraneous root. If you plug y=9 back
into the original equation, you will see that the equation DOES hold true,
so that one is your answer.
Good Luck,
tutor_paul@yahoo.com
Answer by Edwin McCravy(20056)  (Show Source):
You can put this solution on YOUR website!
What is an extraneous solutions?
It's a "phony" solution that you get sometimes,
Even though you haven't made a mistake at all, it
does not check when substituted into the original
equation.
Tutor Paul above gave an example of a radical
equation which has an extraneous solution. However,
when you study extraneous solutions for the first
time, most likely you haven't studied radical
equations yet, but only rational equations. So
I think these examples would be more in line
with what you are studying in school. I'm
guessing Paul's examples above are more
advanced that where you are in the course at the
present time, so I give these 2 examples:
Example 1:
When you clear of fractions and solve the
equation, you get the "solution" x = 1
However x = 1 does not check because it
results in a denominator being 0, which
is undefined. So the equation has no
solution at all! x = 1 is called an
EXTRANEOUS solution, which is really
not a solution at all.
Example 2:
When you multiply through by the LCD
and solve the resulting quadratic
equation, you get solutions x=2 and
x=1. However when we try to check the
solution x=2, it causes the first
and last denominators to become 0, which
is undefined. However x=1 checks. So
we discard the extraneous solution x=2
and keep the x=1 as the only solution.
Edwin
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