When does an equation involving rational expressions have straneous root

Many mathematical operations are reversible, meaning you can go not just from A to B, but by the same reason go from B back to A.

Example x + 4 = 7. Subtract 4 from each side to get x = 7 - 4 = 3. But you could, if you wished, go from x = 3 to x + 4 = 7 by simply doing the opposite, adding 4.

Now consider an operation that is NOT reversible.

Example: Given sqrt(x) = x - 2. Square each side to get x = (x - 2)^2 =x^2 - 4x + 4, or

x^2 - 5x + 4 = 0. Factoring gives (x - 1) (x - 4) = 0. So, IF the original equation holds for a real number x, then the only possibilities are x = 1, x = 4.

If you plug x = 4 back into original equation, you get sqrt(4) = 4 - 2, which is true. But, if you plug in x = 1, you get sqrt(1) = 1 - 2, which is NOT true, as sqrt(1) = 1, not -1. The logic of the derivation was that, IF original equation were true, then x could possibly be only 1 and/or 4. In effect, it was really telling you what x is NOT, namely any number besides 1 and 4. Once you test the two “candidates” you see that one works and the other doesn’t.

Why does this happen? It has to do with the fact that square root is only a principal branch. Just as sqrt(16) is only 4 and not -4, so we are dealing with a limited possibility with which to start. Squaring glosses over this, and when you solve a quadratic equation, it generally gives two solutions (not nec. real). Because sqrt(x) started out to be nonnegative, that created the problem at end.