Common Types of Mistakes: Extraneous Solutions
Sometimes a method used to solve an equation gives extraneous solutions - values which arise from the solution method but that are not solutions of the original equation. This can happen when a step in the solution process is not strictly reversible, for example, if both sides of an equation are squared at some point.
Example 1: Solve
Solving this equation leads (initially) to two solutions. In the first step both sides of the equation are squared.
Note that both x=1 and x=-2 are solutions of the equation x2=2-x, the equation obtained after squaring both sides. However, only x=1 is a solution of the original equation. The other value arising from the method of solution, x=-2, is not a solution of the original equation. It is a solution of the equation
We call x=-2 an extraneous solution. To avoid incorrectly reporting extraneous solutions to equations you should check whether or not each value that arises from the solution method is a solution by substituting that value into the original equation. It's expecially important to do this when, for example, both sides of an equation were squared sometime during the solution method.
Sometimes extraneous solutions arise purely from domain considerations.
Example 2: Solve
The first step in solving this equation is to undo the natural logarithms by applying the exponential function to both sides.
While the equation x2-6=x does indeed have these two solutions, x=-2 is not a solution of the original equation because ln(-2) does not exist (as a real number, at least). Thus x=-2 is an extraneous solution.
The domain can also be a reason to reject extraneous solution in applied ("word") problems in which a solution must (for example) be positive. If you solve an equation arising in a problem asking (say) for the lengths of the sides of a pig pen that maximizes the area of the pen with a limited amount of fence material, you can reject any negative length possible answers right away.
Mistakes arising from extraneous solutions for equations can be subtle. Know to look for them in situations like those mentioned on this page.
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