An extraneous solution is a value that satisfies an equation obtained while solving an original equation but does not satisfy the original equation. It can be introduced by an operation that is not reversible for every value, such as squaring both sides, multiplying by an expression that can vanish, or applying a function that is not an injection.
For example, squaring
 |
gives 
,
whose solutions are 
and 
.
Substitution into the original equation shows that 
is a solution, while 
is an extraneous solution because the principal
square root is nonnegative.
Candidate solutions obtained after a potentially nonreversible step must therefore be checked in the original equation. An extraneous solution is not the same as a value excluded from the domain; either circumstance can cause a candidate to fail the original equation.
