Extraneous Solutions – Part 1 of 3?

Disclaimer

Within my small inner circle of math teachers, the mystery of extraneous solutions seems to be the issue of the year. I have so much to say on this topic (algebraic, logical, pedagogical, historical, linguistic) that I don’t really know where to begin. My only disclaimer is that I’m not really sure if this topic is all that important.

Solving an Equation with a Radical Expression

Consider the following equation:

(1) Image may be NSFW.
Clik here to view.

One hardly needs algebra skills or prior knowledge to solve this, but prior experience suggests trying to isolate Image may be NSFW.
Clik here to view..

(2) Image may be NSFW.
Clik here to view. (we subtract 5 from both sides)

(3) Image may be NSFW.
Clik here to view. (we divide both sides by 2)

Now, if the square root of something is 3, then that something must be 9, so it immediately follows that

(4) Image may be NSFW.
Clik here to view.

(5) Image may be NSFW.
Clik here to view. (we subtract 8 from both sides)

Squaring Both Sides

In my transition from (3) to (4), I used a bit of reasoning. Some conversational common sense told me that “if the square root of something is 3, then that something must be 9”. But that logic is usually just reduced to an algebraic procedure: “squaring both sides”. If we square both sides of equation (3), we get equation (4).

On the one hand, this seems like a natural move. Since the meaning of Image may be NSFW.
Clik here to view. is “the (positive) quantity which when squared is Image may be NSFW.
Clik here to view.“, the expression Image may be NSFW.
Clik here to view. is practically begging us to square it. Only then can we recover what lies inside. A quantity “which when squared is Image may be NSFW.
Clik here to view.” is like a genie “which when summoned will grant three wishes”. In both cases you know exactly what to do next.

Unfortunately, squaring both sides of an equation is problematic. If Image may be NSFW.
Clik here to view. is true, then Image may be NSFW.
Clik here to view. is also true. But the converse does not hold. If Image may be NSFW.
Clik here to view., we cannot conclude that Image may be NSFW.
Clik here to view., because opposites have the same square.

This leads to problems when solving an equation if one squares both sides indiscriminately.

A Silly Equation Leads to Extraneous Solutions

Consider the equation,

(6) Image may be NSFW.
Clik here to view.

This is an equation with one free variable. It’s a statement, but it’s a statement whose truth is impossible to determine. So it’s not quite a proposition. Logicians would call it a predicate. Linguistically, it’s comparable to a sentence with an unresolved anaphor. If someone begins a conversation with the sentence “He is 4 years old”, then without context we can’t process it. Depending on who “he” refers to, the sentence may be true or false. The goal of solving an equation is to find the solution set, the set of all values for the free variable(s) which make the sentence true.

Equation (6) is only true if Image may be NSFW.
Clik here to view. has value 4. So the solution set is Image may be NSFW.
Clik here to view.. But if we square both sides for some reason…

(7) Image may be NSFW.
Clik here to view. has solution set Image may be NSFW.
Clik here to view.

We began with Image may be NSFW.
Clik here to view., “did some algebra”, and ended up with Image may be NSFW.
Clik here to view.. By inspection, Image may be NSFW.
Clik here to view. is a solution to Image may be NSFW.
Clik here to view., but not to the original equation which we were solving, so we call Image may be NSFW.
Clik here to view. an “extraneous solution”. [Extraneous – irrelevant or unrelated to the subject being dealt with]

Note that the appearance of the extraneous solution in the algebra of (6)-(7) did not involve the square root operation at all. But this example was also a bit silly because no one would square both sides when presented with equation (6), so let’s look at a slightly less silly example.

Another Radical Equation

(8) Image may be NSFW.
Clik here to view.

(9) Image may be NSFW.
Clik here to view.

(10) Image may be NSFW.
Clik here to view.

People paying attention might stop here and conclude (correctly) that (10) has no solutions, since the square root of a number can not be negative. Closer inspection of the logic of the algebraic operations in (8)-(10) enables us to conclude that the original equation (8) has no solutions either. Since Image may be NSFW.
Clik here to view., any solution to (8) will also be a solution to (9) and vice versa. Since Image may be NSFW.
Clik here to view., any solution to (9) will also be a solution to (10) and vice versa. So equations (8), (9), and (10) are all “equivalent” in the sense that they have the same solution set.

But what if the equation solver does not notice this fact about (10) and decides to square both sides to get at that information hidden inside the square root?

(11) Image may be NSFW.
Clik here to view.

(12) Image may be NSFW.
Clik here to view.

Again we have an extraneous solution. Image may be NSFW.
Clik here to view. is a solution to (12), but not to the original equation (8). Where did everything go wrong? By the previous logic, (8), (9), and (10) are all equivalent. (11) and (12) are also equivalent. So the extraneous solution somehow arose in the transition from (10) to (11), by squaring both sides.

So unlike subtracting 5 from both sides or dividing both sides by 2, squaring both sides is not an equivalence-preserving operation. But we tolerate this operation because the implication goes in the direction that matters. If Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view., so if Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are expressions containing a free variable Image may be NSFW.
Clik here to view., any value of Image may be NSFW.
Clik here to view. that makes Image may be NSFW.
Clik here to view. true will also make Image may be NSFW.
Clik here to view. true.

In other words, squaring both sides can only enlarge the solution set. So if one is vigilant when squaring both sides to the possible creation of extraneous solutions, and is willing to test solutions to the terminal equation back into the original equation, the process of squaring both sides is innocent and unproblematic.

Those Who are Still Not Satisfied

Still there are some who are not satisfied with this explanation: “Why does this happen? What is really going on? Where do the extraneous solutions come from? What do they mean?”

One source of the problem is the square root operation itself. Image may be NSFW.
Clik here to view. is, by the conventional definition, the positive quantity which when squared is Image may be NSFW.
Clik here to view.. The reason that we have to stress the positive quantity is that there are always two real numbers that when squared equal any given positive real number. There are a few slightly different ways of making this same point. The operation of squaring a number erases the evidence of whether that number was positive or negative, so information is lost and we are not able to reverse the squaring process.

We can also phrase the phenomenon in the language of functions. Since squaring is a common and useful mathematical practice, information will often come to us squared and we’ll need an un-squaring process to unpack that information. Image may be NSFW.
Clik here to view., for all the reasons just mentioned, is not a one-to-one function, so strictly speaking, it is not invertible. But un-squaring is too important, so we persevere. As with all non-one-to-one functions, we first restrict the domain of Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. to make it one-to-one. This inverse, Image may be NSFW.
Clik here to view. thus has a positive range and so the convention that Image may be NSFW.
Clik here to view. is born. So every use of the square root symbol comes with the proviso that we mean the positive root, not the negative root. We inevitably lose track of this information when squaring both sides.

[Note: Students can easily lose track of these conventions. After a lot of practice solving quadratic equations, moving from Image may be NSFW.
Clik here to view. effortlessly to Image may be NSFW.
Clik here to view., students will often start to report that Image may be NSFW.
Clik here to view..]

The convention that we choose the positive root is totally arbitrary. In a world in which we restricted the domain of  Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. before inverting, Image may be NSFW.
Clik here to view. would be Image may be NSFW.
Clik here to view.. In that world, Image may be NSFW.
Clik here to view. is a perfectly good solution to Image may be NSFW.
Clik here to view., not extraneous at all.

A Trigonometric Equation which Yields an Extraneous Solution

For parallelism, consider the (somewhat artificial) equation:

(13) Image may be NSFW.
Clik here to view.

Like in (10), careful and observant solvers might notice that the range of the Image may be NSFW.
Clik here to view. function is Image may be NSFW.
Clik here to view. and correctly conclude that the equation has no solutions. But there seems to be a lot going on inside that Image may be NSFW.
Clik here to view. expression, so many will rush ahead and try to unpack it by “cosineing”. Indeed, since Image may be NSFW.
Clik here to view., this seems innocent.

(14) Image may be NSFW.
Clik here to view.

(15) Image may be NSFW.
Clik here to view.

(16) Image may be NSFW.
Clik here to view.

But Image may be NSFW.
Clik here to view. is an extraneous solution since Image may be NSFW.
Clik here to view. not Image may be NSFW.
Clik here to view..

The explanation for this extraneous solution will be similar to the logic we used above. If Image may be NSFW.
Clik here to view., then Image may be NSFW.
Clik here to view., so if Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are expressions containing a free variable Image may be NSFW.
Clik here to view., any value of Image may be NSFW.
Clik here to view. that makes Image may be NSFW.
Clik here to view. true will also make Image may be NSFW.
Clik here to view. true. So we will not lose any solutions by “taking the cosine of both sides”. But as the cosine function is not one-to-one, Image may be NSFW.
Clik here to view. does not imply that Image may be NSFW.
Clik here to view.. So taking the cosine of both sides, just like squaring both sides, can enlarge the solution set.

The above paragraph explains why extraneous solutions could appear in the solution of (13), but maybe not why they do appear. For that, we again must look to the presence of the Image may be NSFW.
Clik here to view. function. Since Image may be NSFW.
Clik here to view. is not one-to-one, we had to arbitrarily restrict its domain to Image may be NSFW.
Clik here to view. prior to inverting. So every use of the Image may be NSFW.
Clik here to view. symbol comes with its own proviso that we are referring to a number in a particular interval of values. In a world in which we had restricted the domain of Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. prior to inverting, Image may be NSFW.
Clik here to view. would be a perfectly good solution to Image may be NSFW.
Clik here to view., not extraneous at all.

The above examples seem to suggest that one can avoid dealing with extraneous solutions by carefully examining one’s equations at each step. But in practice, this really isn’t possible. I saved the fun examples for the end, but as this post is already way way too long, they will have to wait for a bit later.

-Will Rose

Thanks

Thanks to John Chase for letting me guest post on his blog. Thanks to James Key for encouraging me again and again to think about extraneous solutions.