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.
Read more about this topic: Extraneous And Missing Solutions
Famous quotes containing the words missing and/or division:
“What’s missing is the eyeballs
in each of us, but it doesn’t matter
because you’ve got the bucks, the bucks, the bucks.”
—Anne Sexton (1928–1974)
“Affection, indulgence, and humor alike are powerless against the instinct of children to rebel. It is essential to their minds and their wills as exercise is to their bodies. If they have no reasons, they will invent them, like nations bound on war. It is hard to imagine families limp enough always to be at peace. Wherever there is character there will be conflict. The best that children and parents can hope for is that the wounds of their conflict may not be too deep or too lasting.”
—New York State Division of Youth Newsletter (20th century)