Extraneous and Missing Solutions - Extraneous Solutions: Multiplication

Extraneous Solutions: Multiplication

One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions. However, strictly speaking, this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before. For example, consider the following simple equation:

If we multiply both sides by zero, we get:

This is true for all values of x, so the solution set is all real numbers. But clearly not all real numbers are solutions to the original equation. The problem is that multiplication by zero is not invertible: if we multiply by any nonzero value, we can undo it immediately by dividing by the same value, but division by zero is not allowed, so multiplication by zero cannot be undone.

More subtly, suppose we take the same equation and multiply both sides by x. We get:

This quadratic equation has two solutions, −2 and 0. But if zero is substituted for x into the original equation, the result is the invalid equation 2 = 0. This counterintuitive result occurs because in the case where x=0, multiplying both sides by x multiplies both sides by zero, and so necessarily produces a true equation just as in the first example.

In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero. But it's not sufficient to exclude these values, because they may have been legitimate solutions to the original equation. For example, suppose we multiply both sides of our original equation x + 2 = 0 by x + 2. We get:

This quadratic equation has only one real solution: x = −2, and this is a solution to the original equation, so it cannot be excluded, even though x + 2 is zero for this value of x.