Extraneous and Missing Solutions - Missing Solutions: Division

Missing Solutions: Division

Extraneous solutions are not too difficult to deal with because they just require checking all solutions for validity. However, more insidious are missing solutions, which can occur when performing operations on expressions that are invalid for certain values of those expressions.

For example, if we were solving this equation, the correct solution is to subtract 4 from both sides, then divide both sides by 2:

By analogy, we might suppose we can solve the following equation by subtracting 2x from both sides, then dividing by x:

The solution x = −2 is in fact a valid solution to the original equation; but the other solution, x = 0, has disappeared. The problem is that we divided both sides by x, which is zero when x = 0.

It's generally possible (and advisable) to avoid dividing by any expression that can be zero; however, where this is necessary, it's sufficient to ensure that any values of the variables that make it zero also fail to satisfy the original equation. For example, suppose we have this equation:

It is valid to divide both sides by x−2, obtaining the following equation:

This is valid because the only value of x that makes x−2 equal to zero is x=2, and x=2 is not a solution to the original equation.

In some cases we're not interested in certain solutions; for example, we may only want solutions where x is positive. In this case it's okay to divide by an expression that is only zero when x is negative, because this can only remove solutions we don't care about.