If x^2 + y^2 = 29, what is the value of (x - y)^2 ? (1) xy = 10 (2) x

SOLUTION

If x^2 + y^2 = 29, what is the value of (x - y)^2 ?

Given: \(x^2+y^2=29\).
Question: \((x-y)^2=x^2-2xy+y^2=(x^2+y^2)-2xy=29-2xy=?\) So, basically we should find the value of \(xy\)

(1) xy = 10. Directly gives us the value we needed. Sufficient.

(2) x = 5. Now even if we substitute the value of \(x\) in \(x^2+y^2=29\) we'll get two values for \(y\): 2 and -2, hence two values for \((x-y)^2\): 9 and 49. Not sufficient.

Answer: A.
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If x^2+y^2=29, what is the value of (x-y)^2?
(1) xy=10
(2) x=5

The OA says the answer is A whereas I think statement 2 is also sufficient as given below.
x=5 hence y^2=29-25=4-->y=+2 or -2. Hence (x-y)^2= (5-2)^2 or (5+2)^2

Is it because of two answer choices, Statement 2 is insufficient?

Precisely! When the question is asking for the value of something, then "sufficient" means you can determine a unique individual unambiguous value. There is absolutely only one place on the entire infinity of the number line where that value could be --- that's what "sufficient" means. If you can narrow things down to two values, it's not sufficient. Does that make sense?

Mike
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Hi All,

The posters in this thread all seem comfortable with the math involved, so I won't rehash any of that here. Instead, I won't to point out a common-enough design 'element' in DS questions. In many prompts, the question writers are going to test the 'thoroughness' in your thinking. Can you "see" information in more than one way? As such, when dealing with DS questions, you should look for opportunities to "rewrite" information (and sometimes 'rewriting' the question is the "key" to solving it).

Here, we're told that X^2 + Y^2 = 29. That's an interesting piece of information and not something that we would be given all that often. There MUST be a reason why the writer put it there.......

The question asks for the value of (X-Y)^2. This is one of the Classic Quadratics, so it should get you thinking about FOILing. If we just take a moment to FOIL the question, we get....

"What is the value of X^2 -2XY + Y^2?"

NOW the purpose of that original piece of information is clear....it's part of the solution to the question. If we can figure out the value of XY, then we'll have the answer to the question. From here, the rest of the math is pretty straight-forward.

Remember to look for these pattern-based shortcuts. They will appear repeatedly and while they take a little bit of extra work, doing THAT work will make solving the problem that much easier.

GMAT assassins aren't born, they're made,
Rich
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If x^2 + y^2=29, and according to stmt II x=5, then why can we not derive from that that y=2 and xy=10?

Hi Anonamy,

If you plug X=5 into X^2 + Y^2=29, then what does that tell you about the value of Y. Be sure to be THOROUGH here. Is Y positive or negative? How would that impact the answer to the given question? Without too much work, you should be able to PROVE that there is more than one answer to the given question when X=5.

GMAT assassins aren't born, they're made,
Rich
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(x+y)^2 = x^2 + y^2 - 2xy. We have x^2 + y^2 = 29 already, we need to find xy

(1) Exactly what we need: Suff
(2) Can find y^2 but we dont know whether y is positive or not: Insuff
--> Answer A

Since we want the value of \((x-y)^2\), let’s first expand it and write it as \((x-y)^2 = x^2 + y^2 + 2xy\).

We see that the question gives us \(x^2 + y^2\) = 29; so if we are to find the value of \((x-y)^2\), then we need the value of xy.

Statement I alone is sufficient since it gives us the value of xy directly. Note that this is a DS question and you are not expected to plug in the value of xy and \(x^2 + y^2\) into the equation and find out the final value of \((x-y)^2\).

Statement I alone is sufficient to say that we can find the value of \((x-y)^2\). Answer options B, C and E can be eliminated. Possible answer options are A or D.

From statement II alone, x = 5. Now, using this in the equation given as part of the question,\( x^2 + y^2\) = 29, we can say \(y^2\) = 4 or y = 2 or -2.

If x = 5 and y = 2, \((x-y)^2\) = 9; if x = 5 and y = -2, \((x-y)^2\) = 49.

Can we answer the question with a definitive answer? We cannot. Statement II alone is insufficient to find a unique answer to the question. Answer option D can be eliminated.

The correct answer option is A.

Hope that helps!
Aravind B T
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Solution:

We need to determine the value of (x - y)^2 given that x^2 + y^2 = 29. Notice that (x - y)^2 = x^2 + y^2 - 2xy. Since x^2 + y^2 = 29, then (x - y)^2 = 29 - 2xy. Therefore, if we know the value of xy, then we know the value of (x - y)^2.

Statement One Alone:

Since we know xy = 10, we see that (x - y)^2 = 29 - 2xy = 29 - 2(10) = 9. Statement one alone is sufficient.

Statement Two Alone:

Let’s substitute x = 5 in x^2 + y^2 = 29:

5^2 + y^2 = 29

25 + y^2 = 29

y^2 = 4

y = 2 or -2

If y = 2, then (x - y)^2 = (5 - 2)^2 = 3^2 = 9. If y = -2, then (x - y)^2 = (5 - (-2))^2 = 7^2 = 49. We see that we have two possible values for (x - y)^2. Statement two alone is not sufficient.

Answer: A
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If x^2 + y^2 = 29, what is the value of (x - y)^2 ?

Before we start analyzing statements lets translate the quadratic: x^2 -2xy+ y^2 =?

Wait a second! Look at the given information! We can now break this equation down to -2xy + 29 = ?

Now we just need -2xy!

(1) xy = 10

Suff. This gives us EXACTLY what we need.

(2) x = 5

My brain wants to immediately say that this is sufficient, but lets investigate and plug 5 into the given equation:

5^2 + y^2 = 29

So Y now contains two values (2 and -2)

Because Y is not squared in -2XY We need an EXACT value of Y and will end up with two different products and therefore answers

-2*5*-2=20 20+29=49

-2*5*2=-20 -20+29=9

This answer is Not Suff.

Choose A.

Nice Problem.

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