What is the value of x² + y²?
(1) x + y = 4
(2) xy = 2

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(C) If we use statement (1) by itself, we can solve it for x. x = 4 – y. The desired sum, x² + y², equals (4 – y)² + y². We do NOT know the value of y. Therefore we do NOT have a definite value of the desired sum. Statement (1) by itself is NOT sufficient.

If we use statement (2) by itself, it implements that y can NOT be 0 and we can solve it for x. x = 2/y. The desired sum, x² + y², is (2/y)² + y². We do NOT know the value of y. Therefore we do NOT have a definite value of the desired sum. Statement (2) by itself is NOT sufficient.

If we use the both statements together, let’s manipulate them a little. First, multiply the second one by 2 to get 2xy = 4. Then square the first one. x² + 2xy + y² = 4². Subtract the second one from the first one to get x² + y² = 16 – 4. Therefore the desired sum equals 16 – 4 = 12. Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C.
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It seems to me that (2) would be correct. If we're looking to find the value of x² + y², all we would need to know is "xy" because:
x² + y² = x² + 2xy + y²
so: x² + y² = -2xy
if xy=2, then -2xy= (-2)2= -4
therefore x² + y² =(-4)

The result you got is impossible, because the sum of two squares must be a non-negative number. Therefore there must be a mistake in your reasoning.

Here it is:The first equality is true only if one of the variables is 0 because it can be simplified into 0 = 2xy. Probably, you confused it with (x + y)² = x² + 2xy + y², which is true for any values.

The second equality does NOT come from the first one. It is the same as x² + 2xy + y² = 0, which is (x + y)² = 0.
That is NOT implied by anything in the question statement.

No. (x + y)(x + y) = (x + y)²
We can either use the formula for (x + y)² or multiply (x + y)(x + y) to see that

(x + y)(x + y) = x² + 2xy + y²


Here are the two sets of values that fit Statement (1), but yield different results:
x = y = 2
(1) x + y = 4 holds true: 2 + 2 = 4
In this case x² + y² = 2² + 2² = 4 + 4 = 8

x = 1; y = 3
(1) x + y = 4 holds true: 1 + 3 = 4
In this case x² + y² = 1² + 3² = 1 + 9 = 10

There are two possible variants we can find from the system of the equations (1) and (2):
x = 2 + √2 and y = 2 – √2
or
x = 2 – √2 and y = 2 + √2

In any of the above variants
xy = (2 + √2)(2 – √2) = 2² – (√2)² = 4 – 2 = 2

x + y = (2 + √2) + (2 – √2) = 4

Note, that you do not need to find the actual values of x and y in this question. (Under the assumption that statements in a data sufficiency problem can not contradict each other. So (1) and (2) must define at least one value of (x, y) or more)