Question

What are extraneous solutions of an equation? Why do they sometimes occur in the process of solving rational or radical equations?

Answer 1

an extraneous solution is a solution that doesn't solve the original problem. they can come about in several ways. one way is if you're looking for the length of something and one of the solutions is negative. negative lengths are not permissible so the negative answer is extraneous. it's a solution to the immediate equation that generated it, but it's not a solution to the original problem. other ways to get extraneous solutions are: 1. the value makes the denominator in the original equation equal to 0 which is not permissible. 2. the value makes the radicand in a radical expression negative when the root is even. example: square root of -1 is not permissible. cube root of -1 is. 4th root of -1 isn't, etc. 4th root of -1 is equal to the square root of the square root of -1 which we already know is not permissible. number 1 is involved in a rational equation. number 2 is involved in a radical equation. -----