Because an invalid operation has been performed. Two of the most common examples are multiplication by zero (to remove fractions) and squaring both sides of an equation (to remove radicals). These procedures result in extraneous solutions.

Extraneous solutions could be avoided by the use of correct procedures, but they are not taught, as it is thought to be easier to check the answers at the end.

Examples:

Incorrect: (P/Q = 0) ≡ (P = 0)

Correct: (P/Q = 0) ≡ (P = 0 and Q ≠ 0)

Incorrect: (√P = Q) ≡ (P = Q^2)

Correct: (√P= Q) ≡ (P = Q^2 and Q ≥ 0)