Extraneous Solutions

Question

How do problems with extraneous solutions differ from problems with no real solutions or no solution at all? How are they similar? 

Answer 1

Extraneous solutions appear to be valid solutions, but when checked back into the original equation they give a false result. You must exclude extraneous solutions from your answer.

No solution means there is no value that can make the statement true. Ex. parallel lines have no solution because they never intersect.

No real solution implies that there could possibly be imaginary solutions.

Answer 2

Extraneous solutions are created be using certain methods to solve the problem. They are not, never were, or never will be solutions. They are created by the process that is used to solve. For example, if I have the equation x - 1 = 0. It is clear that one method to solve this is to add 1 to both sides of the equation to get x = 1. But what if I decide to multiply both sides of the equation by zero (which is a legitimate operation in Algebra). I would get 0 = 0 and that is true for any x so my solution (instead of x = 1) would be x = all real numbers. I created extraneous solutions by the process I used to solve the problem.

Problems with no solution are completely different. These problems, as noted, do not have a solution.