Lesson 3

The extraneous root in the previous example was x = −6. You may have verified this by substituting x = −6 into the original equation, as shown below.

 

 

Another way that you may have confirmed that x = −6 is extraneous is to consider the domain of the function. The expression  can be expressed using piecewise notation as

 

 

In Case 1, where x − 6 > 0, since x = −6 is not in the domain of x ≥ 6, this solution is an extraneous one.

 

In Case 2, where x − 6 < 0, since x = 2 was in the domain x < 6, this solution is valid.

 

In the future you may wish to substitute values into the original equation to verify their validity. Alternatively, you may wish to compare obtained solutions to the appropriate domain to check for extraneous solutions.

 

Self-Check 3