extraneous solution

An extraneous solution is a number that looks like a solution — it pops out of your algebra — but does not actually satisfy the original equation. It is a false positive: the steps were legal, yet the answer they produced was never really a solution.

Extraneous solutions appear whenever a solving step is not perfectly reversible. The two famous sources are multiplying both sides by an expression containing the variable (the move used to clear denominators, which can equal zero for some value) and squaring both sides (used with radical equations, which can create sign mismatches). In rational equations, the extraneous value is typically one of the excluded values.

The cure is simple and mandatory: substitute every candidate back into the original equation and keep only those that genuinely work. Discarding an extraneous solution is not optional bookkeeping — it is the difference between a correct answer and a wrong one.