What is the value of 1/(x^2 - y^2) - 1/(x^2 + 2xy + y^2) ?

Question

What is the value of  \frac{1}{x^2 - y^2} - \frac{1}{x^2 + 2xy + y^2} ?

(1)  2y = x^2 - y^2

(2)  x + y = 4

DangerPenguin · Answer

We can start off by factoring the denominators to get

1 / (x+y)(x-y) - 1 / (x+y)(x+y)

Statement 1: not sufficient
Factor to get 2y=(x+y)(x-y)
Plug into what we were given to get 1/2y - 1 / (x+y)(x+y). Not enough info.

Statement 2: not sufficient
Plug 4 in for (x+y) in the equation we were given originally to get 1/4(x-y) - 1/16. Not enough info.

Down to C and E.

Combined we know that 2y must equal 4(x-y) so we can say that 1/4(x-y) - 1/16 = 1/2y - 1/16. We still don't have a value for y and cannot get a value of the equation.

Answer E!

gillesalex07 · Answer

First, one should notice that 1/(x^2 - y^2) - 1/(x^2 + 2xy + y^2) = 2*y/ .

(1) gives 1/(x^2 - y^2) - 1/(x^2 + 2xy + y^2) = 1/(x+y)
INSUFFICIENT

(2) clearly INSUFFICIENT

(1) + (2) gives 1/(x^2 - y^2) - 1/(x^2 + 2xy + y^2) = 1/4

Answer (C)

devmillenium2k · Answer

Let's Simplify the equation - 1/(x^2 - y^2) - 1/(x^2 + 2xy + y^2)

1/(x+y)(x-y) - 1/(x+y)^2 
Take LCM and solve (x+y) - (x-y) /(x-y) (x + y)^2 => 2y / (x-y) (x + y)^2 ------ (1)

Statement 1 : 2y = x^2 - y^2 => 2y = (x+y) (x-y)
Put this value in equation 1) and check,

(x+y) (x-y) / (x-y) (x+y) (x+y) => 1/ x+y .... Not enough info
 Not Sufficient.  Option A & D are out

Statement 2 : x + y = 4
Put this value in equation 1) and check,
2y / (x-y) (4)^2 => 2y / 16 * (x-y) .... Not enough info
 Not Sufficient.  Option b is out

Combining both the statement,
We already have 1/ (x+y) from statement 1 as solution of equation and statement 2 provides the value for (x + y) as 4
so we can get 1/4 as final solution from both the statements.

Hence answer is C

kinghyts · Answer

1 / x^2-y^2 - 1 / (x+y)^2

=> {x+y -(x-y) } / 
=> 2y/{(x^2-y^2)*(x+y)}

From statement 1, 2y = x^2 - y^2 
we can say that ,
=> 2y /2y *(x+y) = 1/(x+y). Insufficient

From statement 2, x + y = 4
we can say that ,
=> 2y/ 16*(x-y). Insufficient

Combining both, 
we can the value = 1/(x+y) = 1/4

C) should be the answer

Bunuel · Answer

Official Solution:

What is the value of  \frac{1}{x^2 - y^2} - \frac{1}{x^2 + 2xy + y^2} ? 
 
We must determine the value of the expression  \frac{1}{x^2 - y^2} - \frac{1}{x^2 + 2xy + y^2} . When a problem presents an opportunity to factor or find a common denominator, it is usually a good idea to do so.

First, factor the denominator of each fraction. The denominator of the fraction on the left is a difference of squares:  \frac{1}{x^2 - y^2} = \frac{1}{(x + y)(x - y)} . The denominator of the fraction on the right is the expanded form of the quadratic expression  (x + y)^{2} :  \frac{1}{x^2 + 2xy + y^2} = \frac{1}{(x + y)^2} .

Thus, the fraction can be rewritten:  \frac{1}{(x + y)(x - y)} - \frac{1}{(x + y)^2} . The common denominator of these two fractions is  (x + y)^{2}(x - y) . Multiply the first fraction by  \frac{x + y}{x + y}  and the second fraction by  \frac{x - y}{x - y}  and rewrite:  \frac{x + y}{(x + y)^{2}(x - y)} - \frac{x - y}{(x + y)^{2}(x - y)} .

Combine the terms by subtracting:  \frac{x + y - (x - y)}{(x + y)^{2}(x - y)} = \frac{2y}{(x + y)^{2}(x - y)} .

Statement 1 says that  2y = x^2 - y^2 , or  2y = (x + y)(x - y) . Substitute this into the fraction that we derived above:  \frac{2y}{(x + y)^{2}(x - y)} = \frac{(x + y)(x - y)}{(x + y)^{2}(x - y)} . Cancel the factors that appear in both the numerator and the denominator, leaving  \frac{1}{x + y} . Without more information about  x  or  y , we cannot determine the value of this fraction. Statement 1 is NOT sufficient. Eliminate answer choices A and D. The correct answer choice is B, C, or E.

Statement 2 says that  x + y = 4 . In this case, it will be easier to substitute into the expression  \frac{1}{(x + y)(x - y)} - \frac{1}{(x + y)^2} . Doing so gives:  \frac{1}{4(x - y)} - \frac{1}{(4)^2} . Without more information about  x  and  y , however, we cannot determine the value of this expression. Statement 2 is NOT sufficient. Eliminate answer choice B. The correct answer choice is either C or E.

When the statements are taken together, statement 1 allows us to simplify the fraction to  \frac{1}{x + y} , and statement 2 tells us that  x + y = 4 . Substituting, we find:  \frac{1}{x + y} = \frac{1}{4} . Together, the statements are sufficient to answer the question.
 
Answer: C.

KCL5728 · Answer

I am not sure if I am missing a step here ... but I think this question can be solved in 2 seconds, without any calculation at all?

There are two variables (X & Y), and of course, you cannot solve either X or Y with just one equation (i.e., either equation 1 or equation 2 itself would be insufficient).

If we have TWO equations (i.e., equation 1 + equation 2), it is without a doubt X and Y can be solved, and therefore the value of the equation in the question can be answered, no matter what equation it is (as long as they are only composed of variables X and Y).

So it seems to me the answer is clearly C, without any calculation needed.

Any thoughts are welcomed! Let me know if I am missing a step here.

What is the value of 1/(x^2 - y^2) - 1/(x^2 + 2xy + y^2) ?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards