Download What is an extraneous solution? Remember when you solved

What is an extraneous solution?
Remember when you solved radical equations earlier in the course? Often times you solved a
radical equation then later found that the solution actually did not work. An apparent
solution to an algebraic equation that does not check is called an extraneous solution. For
example, let's look at the following problem:
x −1 = x − 7
Notice that after getting rid of the radical and solving, the solution appears to be
x = 10 or x = 5.
But the solution x = 5 does not check. Therefore, x = 10 is the only solution to this
equation, and x = 5 is an extraneous solution.
You will often encounter extraneous solutions when solving logarithmic equation. For
example, consider the logarithmic equation:
log 2 x + log 2 ( x + 2) = 3
log 2 x( x + 2) = 3
Use log properties to write as one log.
23 = x( x + 2)
Rewrite as an exponent.
8 = x2 + 2x
Simplify.
x2 + 2 x − 8 = 0
Set the equation equal to zero.
( x + 4)( x − 2) = 0
Factor.
x + 4 = 0 or x − 2 = 0
Zero Product Property.
x = −4 or x = 2
Solve for x.
Now, when plugging x = −4 back into the ORIGINAL EQUATION we see that we
obtain a negative number inside the logarithm. Since the domain of a logarithm can only
include numbers greater than 0, this indicates that x = −4 is an extraneous solution.
Therefore, the only solution to this logarithmic equation is x = 2.
BE SURE TO ALWAYS CHECK YOUR ANSWERS WHEN SOLVING A
LOGARITHMIC EQUATION!