When do extraneous solutions occur

However, the real world restriction on the solution is that w is the width of a rectangle, and rectangles can't have negative side lengths. Therefore, -8 is an extraneous solution.

Find all two digit numbers such that when the number is squared, and ten times the number is subtracted, the result is Sometimes a problem has fractions, and a number that works out algebraically actually causes a division by zero, which is illegal, immoral, and socially unacceptable.

As a problem writer for math competitions, I like to deliberately create problems in which this happens, because the very best students check their solutions and realize they've got an extraneous solution. Here's a very simple example:. But the process of multiplying by x - 1 has made a fundamental change to the problem; we no longer have a denominator. Each positive number has two real square roots. For example, the square roots of 4 are 2 and However, when we write radical notation, we are, be definition, referring to the principle square root, which is the positive value.

Thus, a solution may be extraneous because it results from using a negative square root instead of the principle square root. Here's an example:. Now, plug 9 into the original equation, and you'll see that it does work. But what happens when you plug in 4? You get a false equation! But why does that happen? Because on the left-hand side, 4 has two square roots, and one of them -2 does work. Students too often miss the fact that when they check a solution, it can fail for two very different reasons: It may fail because it is an extraneous solution that we introduced by our work in which case the check is an essential part of the work itself ; or it may fail just because it is wrong, a result of a mistake in our work.

If it is extraneous , we can just ignore it and say there is no solution, if we found no non-extraneous solutions ; if it is erroneous , we have to go back and fix our work. So when I teach about this, I always show how to distinguish an extraneous solution from an erroneous solution.

Muhammad knew the need to check the solutions, but the check here seemed too hard, since the solutions were ugly expressions. I gave him two pieces of advice that can simplify the checking process. First, since any extraneous solution to a rational equation will fail by making terms undefined, he only needs to check that:.

Their knowledge of extraneous solutions seems to have displaced what they formerly knew, that they themselves are the most common cause of failed checks. Only ignore failed solutions that you know are extraneous.

Here Phinah had several different issues, so I went through the whole process of solving; but at the end before being asked I pointed out how checking works here:. This is typical: If the check fails, see if changing the sign on one or both radicals would make it correct. There are some other things worth discussing that are related to extraneous solutions, but I will just provide links:. Here I offer an alternative which I had never thought of previously that simplifies the check in some cases:.

Here I discussed the opposite issue, where you can lose a valid solution rather than find an invalid one:. We define the radical as representing only one value so that it is a function , which makes algebra easier.

Your email address will not be published. This site uses Akismet to reduce spam. Learn how your comment data is processed. Skip to content The other day a student I was helping face to face asked how she can know when to check for extraneous solutions of an equation. What causes extraneous solutions? Here is a question that will serve as a good overview. Sarah in asked, Extraneous Routes I know that radical, log, absolute value, and fractional operations sometimes introduce extraneous roots.

When else do I need to check for them? Is it only by an equation? I wouldn't want to claim to give a complete list; any time you use a technique you haven't used before, you should determine for yourself whether it falls in this category. So, 0 is an extraneous solution. Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC.

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