SOLUTION: Rationalize the denominator (3-Sqrt(6))/(x-(3+Sqrt(6))) Answer should be (3x-15+6Sqrt(6)-xSqrt(6))/(x^2-6x+3) but i don't see how they are getting that answer. please expla

Algebra ->  Radicals -> SOLUTION: Rationalize the denominator (3-Sqrt(6))/(x-(3+Sqrt(6))) Answer should be (3x-15+6Sqrt(6)-xSqrt(6))/(x^2-6x+3) but i don't see how they are getting that answer. please expla      Log On


   

Question 715790: Rationalize the denominator
(3-Sqrt(6))/(x-(3+Sqrt(6)))
Answer should be (3x-15+6Sqrt(6)-xSqrt(6))/(x^2-6x+3) but i don't see how they are getting that answer. please explain
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%283-sqrt%286%29%29%2F%28x-%283%2Bsqrt%286%29%29%29
First let's rewrite the denominator as additions:
%283-sqrt%286%29%29%2F%28x%2B%28-3%2B%28-sqrt%286%29%29%29%29
The reason we changed to additions is that we can now use the Associative Property to rearrange the grouping:
%283-sqrt%286%29%29%2F%28%28x%2B%28-3%29%29%2B%28-sqrt%286%29%29%29%29

A factoring pattern, %28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2, is often used to rationalize a multiple-term denominator. It gives us a way to to create an expression which has only perfect square terms. The reason we regrouped the denominator was so that we could more easily see the "a" and "b". If we think of the "(x+(-3))" as the "a" and the "-sqrt%286%29" as the "b", the denominator is a+b. To use the pattern to create the perfect squares we must multiply by a-b:

with the a's in red and the b's in blue. The pattern tells us that the denominator works out to be a%5E2-b%5E2. In the numerator we must multiply each term of the first numerator by each term of the second numerator.

We can use another pattern, a%2Bb%29%5E2+=+a%5E2-2ab%2Bb%5E2 to square %28x%2B%28-3%29%29%5E2 quickly. Simplifying...



which is the same as the answer you posted except it has additions instead of subtractions.