If x^2 + y^2 = 100, what is the value of (x - y)^2 ?

Step 1: Analyse Question Stem

We know that \( x^2 + y^2\) = 100.

The value of \((x – y)^2\) is to be calculated.

\((x-y)^2\) = \(x^2\) +\( y^2\) – 2xy, we need the value of xy to be able to calculate the answer since \(x^2 + y^2\) is given as question data.

Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE

Statement 1: x = \(\frac{48 }{ y}\)

Therefore, xy = 48. This is sufficient to calculate the value of \((x-y)^2\).

Statement 1 alone is sufficient. Answer options B, C and E can be eliminated.

Statement 2: \(x^2−y^2\) = −28

From the question, \(x^2 + y^2\) = 100.

These two equations can be solved as simultaneous equations. Doing so, we get,
\(x^2\) = 36 and \(y^2\) = 64

Therefore, x = +6 or -6 and y = +8 or -8.

This means that we can different values for (x-y) and hence different values for \((x-y)^2\).
Statement 2 alone is insufficient. Answer option D can be eliminated.

The correct answer option is A.
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I am a little confused about when we are supposed to consider negative values of square root and when not. I selected option D, thinking we can discard the value of -6 for x. Can someone please clarify?

Note that in the answer above que is solving even exponents.

Even exponents will always result in positive numbers. I don't really know if what you are asking is this, but when dealing with negative bases it is important to pay attention to the positioning of parenthesis.

-2^2 is not equal to (-2)^2, since -2^2 = -1*2^2=-4 and (-2)^2 = -1^2*-2^2 = 4.

Hope it helps.

Hi, could anyone explain why the first statement is sufficient, please? xy=48 can be 6x8=48 but also (-6)x(-8)=48, therefore I would argue, that this statement is insufficient, even though both possible values for x and y yield the same result of (x-y)^2. Thanks.

Hi,
Because what we care to answer is x*y, not x or y separately. Sufficiency means you have sufficient data to answer the question, not find the exact value of any variables of the question.