Extraneous and missing solutions

In mathematics, an extraneous solution represents a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the original problem. A missing solution is a solution that was a valid solution to the original problem, but disappeared during the process of solving the problem. Both are frequently the consequence of performing operations that are not invertible for some or all values of the variables, which disturbs the chain of logical implications in the proof.

Introduction

Extraneous solutions: multiplication

One of the basic principles of algebra is that one can multiply both sides of an equation by the same expression without changing the equation's solutions. However, strictly speaking this is not true, in that multiplication by certain expressions may introduce new solutions that were not present before. For example, consider the following simple equation:

:$x+2=0$

If we multiply both sides by zero, we get:

:$0=0$

This is true for all values of "x", so the solution set is all real numbers. But clearly not all real numbers are solutions to the original equation. The problem is that multiplication by zero is not "invertible": if we multiply by any nonzero value, we can undo it immediately by dividing by the same value, but division by zero is not allowed, so multiplication by zero cannot be undone.

More subtly, suppose we take the same equation and multiply both sides by "x". We get:

:$x^2+2x=0$

This quadratic equation has two solutions, −2 and 0. But if zero is substituted for "x" into the original equation, the result is the invalid equation 2 = 0. This counterintuitive result occurs because in the case where "x"=0, multiplying both sides by "x" multiplies both sides by zero, and so necessarily produces a true equation just as in the first example.

In general, whenever we multiply both sides of an equation by an expression involving variables, we introduce extraneous solutions wherever that expression is equal to zero. But it's not sufficient to exclude these values, because they may have been legitimate solutions to the original equation. For example, suppose we multiply both sides of our original equation "x" + 2 = 0 by "x" + 2. We get:

:$x^2+4x+4\; =\; 0$

This quadratic equation has only one real solution: x = −2, and this is a solution to the original equation, so it cannot be excluded, even though "x" + 2 is zero for this value of "x".

Extraneous solutions can arise naturally in problems involving fractions with variables in the denominator. For example, consider this equation:

:$frac\{1\}\{x\; -\; 2\}\; =\; frac\{3\}\{x\; +\; 2\}\; -\; frac\{6x\}\{(x\; -\; 2)(x\; +\; 2)\},.$

To begin solving, we multiply each side of the equation by the least common denominator of all the fractions contained in the equation. In this case, the LCD is $(x\; -\; 2)(x\; +\; 2)$. After performing these operations, the fractions are eliminated, and the equation becomes:

:$x\; +\; 2\; =\; 3(x\; -\; 2)\; -\; 6x,.$

Solving this yields the single solution "x" = −2. However, when we substitute the solution back into the original equation, we obtain:

:$frac\{1\}\{-2\; -\; 2\}\; =\; frac\{3\}\{-2\; +\; 2\}\; -\; frac\{6-2\}\{(-2\; -\; 2)(-2\; +\; 2)\},.$

The equation then becomes:

:$frac\{1\}\{-4\}\; =\; frac\{3\}\{0\}\; -\; frac\{12\}\{0\},.$

This equation is not valid, since one cannot divide by zero.

Because of this, the only effective way to deal with multiplication by expressions involving variables is to substitute each of the solutions obtained into the original equation and confirm that this yields a valid equation. After discarding solutions that yield an invalid equation, we will have the correct set of solutions. Note that in some cases all solutions may be discarded, in which case the original equation has no solution.

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:

:$2x+4=0$:$2x=-4$:$x=-2$

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

:$x^2+2x=0$:$x^2=-2x$:$x=-2$

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:

:$x+2=0$

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

:$frac\{x+2\}\{x-2\}=0$

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.

Other operations

Multiplication and division are not the only operations that can modify the solution set. For example, take the problem:

:$sqrt\{x\}\; =\; -1,.$

If we square both sides and simplify, we get:

:$(sqrt\{x\})^2\; =\; (-1)^2,.$:$x\; =\; 1,.$

But "x" = 1 is not a solution of the original equation. Squaring introduced a solution to an equation that has no solutions.

Similarly, suppose we have this equation:

:$x^2\; =\; 4.$

If we take the positive square root of both sides, we get:

:$x\; =\; 2.$

We're not taking the square root of any negative values here, since both "x"² and 4 are necessarily positive. But we've lost the solution "x" = −2. The reason is that "x" is actually not in general the "positive" square root of "x"². If "x" is negative, the positive square root of "x"² is "-x". If the step is taken correctly, it leads instead to the equation:

:$|x|\; =\; 2.$

This equation has the same two solutions as the original one: "x" = 2, and "x" = −2.

Finally, some operations can remove and add solutions at the same time. For example, if we have:

:$x\; =\; -2$

And we take the absolute value of both sides, we get:

:$x\; =\; 2$.

The valid solution "x" = −2 is gone, and the extraneous solution "x" = 2 has appeared. But as in the previous example, in general the absolute value of "x" is not "x". The correct next equation is also here:

:$|x|\; =\; 2.$

This equation has two solutions: "x" = 2, and "x" = −2. The first one is extraneous.