Expert: Scott A Wilson - 11/11/2009

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

Answer
Extraneous solutions arise when we solve the problem by multiplying by some expression that cancels a term in the denominator.  Once the problem has been solved, an extra solution is gotten that really would make the denominator 0, and hence is not a solution.

They might also occur when a squareroot is involed.  Squareroots can be both positive and negative, but if time is involved, only the positive value is being looked for.  We could have a problem like firing something from a 10 ledge and seeing how long it takes to hit the ground.  According to physics, the involves solving a quadratic equation for time.  It gives us two solutions, one positive and one negative.  The negative one is of no interest.  The negative one does have a meaning, though, but we don't really care about that.  What the negative answer would tell us is if the object were fired from the ground at the same speed at which it hits the ground at the other end, it would be 10 high right now.