SOLUTION: Dear Sir/Madam, I am confronted with the following problem: "Rationalize the denominator: ({{{ sqrt( x ) }}} + 5) / ({{{ sqrt( x ) }}} - 5)" In order to solve this problem

Algebra ->  Coordinate Systems and Linear Equations -> SOLUTION: Dear Sir/Madam, I am confronted with the following problem: "Rationalize the denominator: ({{{ sqrt( x ) }}} + 5) / ({{{ sqrt( x ) }}} - 5)" In order to solve this problem      Log On


   

Question 6482: Dear Sir/Madam,
I am confronted with the following problem:
"Rationalize the denominator: (+sqrt%28+x+%29+ + 5) / (+sqrt%28+x+%29+ - 5)"
In order to solve this problem I squared the entire fraction, giving me:
(x+10+sqrt%28+x+%29++25) / (x-10+sqrt%28+x+%29++25)
but that is not the right answer. What am I doing wrong?
Thank you in advance.
Regards,
Mike
Answer by rapaljer(4671) About Me  (Show Source):
You can put this solution on YOUR website!
"Rationalize the denominator: %28sqrt%28x%29+%2B+5%29+%2F%28sqrt%28x%29+-+5%29+"

First of all, DON'T square the entire problem. That changes the problem and the answer will NOT be the same as the problem was. To do this is blasphemous!! You rationalize a denominator by multiplying numerator and denominator by what is called the "conjugate" of the denominator--that is, the same as the denominator with the opposite sign in the middle. In this case, you multiply numerator and denominator by +%28sqrt%28x%29%2B5%29+. The reason we do this?? Because it works!! It eliminates the radicals from the denominator. By the way, it is purely a coincidence that this happens to be the same as the numerator.



When you multiply out the numerator and denominator, after considerable work, you get:


This simplifies to:
%28x+%2B10%2Asqrt%28x%29%2B25%29%2F%28x-25%29

You might also leave this answer without multiplying out the numerator as:
+%28%28sqrt%28x%29+%2B5%29%5E2%29+%2F%28x-25%29+

These fractions are pretty ugly, but they do not reduce. The problem said to rationalize the denominator, which means to eliminate the radicals from the denominator. You did that, so probably either of these will be your final answer. Sometimes in math you just have to get used to ugly answers.

R^2 at SCC