What is a extraneous solution?

Answer 1

A solution to an equation that you get at the end of whatever method you use that does not actually solve the original equation. One well-known example:

1=2 ====>0=0 Therefore, one equals two.

x0 x0

The laws of algebra says that we can do this because we multiplied both sides by zero. Logically, we all know this isn't actually true. This is what extraneous solutions look like when solving linear equations:

2x+3=9 If you assume x=1... 2(1)+3=9 ...and multiply everything by 0...

0=0. Therefore, my guess is correct and x=1. <==This is extraneous

This is a minor flaw in algebra, but not like what Russel's paradox in set theory because there are explanations why this happens. Here's a more useful example of how extraneous roots occur naturally:

Solve this the way every algebra student knows, and you get x=-2. Put this back into the equation, however, and you divide by zero. -2 is an extraneous solution in this case. If the only solutions are extraneous, then the equation cannot be solved.

This is why we check rational equations: we want only actual solutions.