If x and y are integers and if (x-y)^2 + 2*y^2 ...

\((x-y)^2 + 2y^2 = 27\)

Since x and y are integers, \((x-y)^2\) and \(y^2\) are both perfect squares.

Since the left side must sum to 27, \(y^2\) must be equal to a perfect square less than 27, yielding the following options for \(y^2\):
0, 1, 4, 9, 16, 25

To yield a sum of 27, one possible option for \(y^2\) is 1, with the result that \((x-y)^2=25\) and that \(x-y=±5\).
If \(y=±1\), only D among the answer choices will yield a viable option for \(x\):
\(x=4\) and \(y=-1\), with result that \(x-y = 4-(-1) = 5\)

If \(x\) and \(y\) are both integers and if \((x-y)^2 + 2(y^2) = 27\), which of the following could be the value of x?

a. -2
b. -1
c. 3
d. 4
e. 5

This question could seem impossible to answer since we have just one equation and two variables. However, we have two additional constraints.

The first is that \(x\) and \(y\) must both be integers.

The second is that the possible values of \(x\) are limited to the set of answer choices.

So, since this is a GMAT Quant question, we'll know that we'll be able to answer it by using those constraints.

Given that \(x\) and \(y\) are integers and that \((x-y)^2 + 2(y^2) = 27\), what we have are one times one perfect square + two times a second perfect square adding up to \(27\). So, one way we can answer this question is to simply consider all the perfect squares that are less than \(27\), identify which ones add up to \(27\) and find a value of \(x\) that works.

It's not a super elegant way of answering the question, but it demonstrates a key GMAT Quant characteristic, which is that GMAT Quant questions are designed to test our skill in simply finding a way to get to an answer. So, if you don't quickly see an elegant mathematical way to answer a question, it's likely that you'll be fine just using a brute force method to get it done.

There are six perfect squares less than \(27\): \(0\), \(1\), \(4\), \(9\), \(16\), \(25\).

Scanning them, we quickly see that only \(0\), \(1\), \(4\), and \(9\) can be doubled since \(2\) x \(16\) and \(2\) x \(25\) are greater than \(27\).

Then, considering possible ways of getting to \(27\), we see that there are only two: \(25 + 2(1) = 27\) and \(9 + 2(9) = 27\).

In the first case, \(y^2 = 1\). So, \(y\) must be \(1\) or \(-1\). Then, for \((x - y)^2\) to equal \(25\), \(x - y\) must be \(5\) or \(-5\).

So, if \(y = 1\), then \(x\) must be \(6\)\(\) or \(-4\). Since neither \(6\) nor \(-4\) are among the answer choices, this path to \(27\) won't work.

If \(y = -1\), then for \(x - y\) to equal \(5\) or \(-5\), x can be \(4\) or \(-6\).

We see that \(4\) is among the answer choices.

So, the correct answer is (D).

__________________
y^2 could also be 9.

Given the answer choices, 9 is not a viable option for \(y^2\) because \(y=±3\) would require that \(x=0\) or \(x=6\), neither of which is among the five options.
I've edited my response to clarify that 1 is one possible option for \(y^2\).
Since this option yields a value for \(x\) that is among the five answer choices, no need to look any further.

Yes, y^2 = 9 does not yield an x value that is among the answer choices, nor was I implying that it does. I was simply pointing out that 1 is not "the only possible option for y^2".

Agreed. As the Wordle bot states when the bot and I come to the same conclusion, "We are as one."

Bunuel, if we pick a value of X from the given option, will that be a long procedure to find out the solution? Also, if we pick a value from the options, is there a way to do a faster calculation?

I wouldn't approach this question by plugging in options. You'll end up with ugly quadratics for y, trying to figure out if there's an integer solution.

\(3y^2-2xy+x^2-27=0\)

1) Reframe the eq as quadaratic in \(y\)
2) Find Discrimiant as function of \(x\)

\(D(x)=4x^2-4*3(x^2-27)= 4(81-2x^2)\)
\(D(x)>=0\) and of form \(N^2\)

\(D(4)=4(81-32)=14^2\)
\(D(x)>=0\) follows that \(-9/rt(2)<=x<=9/rt(2)\)
\(=> -5<=x<=5\)
\(D(x)\) is integer at \(x=0,4,-4,6,-6\)

Tick \(x=4\)

This is one of the hardest problems I have encountered. Is it normal to get a question like this as your first question on the GMAT Focus? It took me way too long to solve this