x and y are positive integers such that x > y. If 2x^1/2 - 2y^1/2 =

MANHATTAN GMAT OFFICIAL SOLUTION:

To solve this problem we can focus on solving for b as a first step:

\(2b(\sqrt{x}-\sqrt{y})=x-y\)

\(2b=\frac{x-y}{\sqrt{x}-\sqrt{y}}\)

Note that it is OK to divide by \(\sqrt{x}-\sqrt{y}\), since x > y , which implies that \(\sqrt{x}-\sqrt{y}\neq{0}\).

We have solved for 2b, but the result does not match any of the answer choices. Most of the choices are not fractions, so we should try to cancel the denominator. Recognizing that x – y is a well disguised “difference of two squares,” we can factor the numerator and denominator:

\(2b=\frac{(\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})}{\sqrt{x}-\sqrt{y}}\)

Cancel \(\sqrt{x}-\sqrt{y}\) in the numerator and denominator to get \(2b = \sqrt{x}+\sqrt{y}\).

The correct answer is B.

Hello,

could you show me the difference of 2 squares? I cannot see it...

x-y= (\(\sqrt{x}\) - \(\sqrt{y}\) )( \(\sqrt{x}\) + \(\sqrt{y}\) )
This is of the same identity as \(a^2 - b^2\)= (a+b)(a-b)

Well I am still missing sth.

We are at: 2b=(x−y) / (√x−√y)
The direrence of squares is this: (x-y)^2 = (x-y)(x+y). But in our case x - y is not raised to the second power..

Then I though you squared the nominator and the denominator, which could create the above mentioned difference of squares, but in the denominator (√x - √y )( √x + √y), and should also create this in the nominator (x-y) (x+y).

So, I still cannot see the difference of squares...

\((a-b)^2=a^2-2ab+b^2\) is square of the difference.

\(a^2 - b^2=(a-b)(a+b)\) is difference of two squares.

In our case we have x - y, which is \((\sqrt{x}-\sqrt{y})(\sqrt{x}+\sqrt{y})=x-y\).

Hope it's clear.

I just focused on the fact that x and y are integers and the fact that x>y.

So I picked x as 9 and y as 4. (Two easiest perfect squares I could think of. I know I could have picked 1 but lets not be too lazy).

Only option B works then

Bunuel, is that correct approach?

We are given that, 2√x - 2√y = (x –y)/2

Or, 2b = (x –y)/(√x - √y)

Or, 2b = (√x)^2 - (√y) ^2/(√x - √y)

Or, 2b =(√x + √y) (B)

Yes, you can use number plugging for this question.

You should be careful though. When plugging numbers, it might happen that two or more choices give "correct" answer. If this happens, just pick some other number(s) and check again these "correct" options only.

Multiplying both sides of the equation by b, we have:

2b√x - 2b√y = x - y

2b(√x - √y) = x - y

2b = (x - y)/(√x - √y)

Although we have an expression for 2b, it’s not one of the answer choices. However, upon closer inspection, x - y is a difference of two squares if we consider x as (√x)^2 and y as (√y)^2. Therefore, x - y = (√x - √y)(√x + √y), and we have:

2b = (√x - √y)(√x + √y) / (√x - √y)

2b = √x + √y

Answer: B

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