SOLUTION: which choice is equivalent to the fraction below when x>or equal to 2? 4/√ (x-2)-√ x A) -2(√ x+√ x-2) B) 2(√ x+√ x-2) C) -2(√ x-

Algebra ->  Square-cubic-other-roots -> SOLUTION: which choice is equivalent to the fraction below when x>or equal to 2? 4/√ (x-2)-√ x A) -2(√ x+√ x-2) B) 2(√ x+√ x-2) C) -2(√ x-      Log On


   

Question 628430: which choice is equivalent to the fraction below when x>or equal to 2?
4/√ (x-2)-√ x

A) -2(√ x+√ x-2)
B) 2(√ x+√ x-2)
C) -2(√ x-√ x-2)
D) 2(√ x-√ x-2)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
4%2F%28sqrt%28x-2%29-sqrt%28x%29%29
Note that none of the answers are fractions. This means that the fraction above must have had its denominator rationalized and that the resulting fraction reduced to the point that there is no longer any fraction. So what we need to do is rationalize the denominator.

Rationalizing a two-term denominator like yours takes advantage of the %28a%2Bb%29%28a-b%29=a%5E2-b%5E2 pattern. This pattern shows us how to take a tw0-term expression (an a+b or an a-b), multiply it by something and get an expression of perfect squares. With your denominator having a "-" between the terms, it will play the role of a-b. So to get the perfect squares we are looking for, we need to multiply by a+b:

On top we use the Distributive Property to multiply. On the bottom the pattern tells us how it will work out:

Simplifying...
%284%2Asqrt%28x-2%29%2B4%2Asqrt%28x%29%29%2F%28%28x-2%29-%28x%29%29
%284%2Asqrt%28x-2%29%2B4%2Asqrt%28x%29%29%2F%28-2%29
%282%282%2Asqrt%28x-2%29%2B4%2Asqrt%28x%29%29%29%2F%28%28-1%29%2A2%29
%28cross%282%29%282%2Asqrt%28x-2%29%2B2%2Asqrt%28x%29%29%29%2F%28%28-1%29%2Across%282%29%29
%282%2Asqrt%28x-2%29%2B2%2Asqrt%28x%29%29%2F%28-1%29
-2%2Asqrt%28x-2%29%2B%28-2%2Asqrt%28x%29%29
This is the simplified expression. The answers given to you are factored. Factoring out -2 we get:
-2%28sqrt%28x-2%29%2Bsqrt%28x%29%29
which is equal to answer "A".