If x^2 - y^2 = 27, what is the value of (x + y)^2 ?

x^2 - y^2 = 27 OR (x+y)(x-y) = 27
We need to find (x+y)^2, for which we need (x+y). Here various cases are possible for (x+y) and (x-y).. Eg, former could be 9 and latter 3 or former could be 3 and latter 9 or former could be -3 and latter -9.

(1) y=3. Substituting this we get
x^2 - 3^2 = 27 OR x^2 = 27+9=36. Two values are possible for x here: 6 and -6, this will give two different answers for x+y. So Insufficient

(2) x-y=3. Substituting this we get
(x+y)*3 = 27 or x+y =9. So we can get (x+y)^2. Sufficient.

Hence B answer

Target question: What is the value of (x+y)²?

Given: x² - y² = 27

Statement 1: y = 3
Take x² - y² = 27 and replace y with 3 to get: x² - (3)² = 27
Evaluate: x² - 9 = 27
Simplify: x² = 36
So, EITHER x = 6 OR x = -6
Let's test each case:
Case a: x = 6 and y = 3. So, (x + y)² = (6 + 3)² = 9² = 81. So, in this case, the answer to the target question is (x+y)² = 81
Case b: x = -6 and y = 3. So, (x + y)² = (-6 + 3)² = (-3)² = 9. So, in this case, the answer to the target question is (x+y)² = 9
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x - y = 3
Take x² - y² = 27 and FACTOR the left side to get: (x + y)(x - y) = 27
Replace (x-y) with 3 to get: (x + y)(3) = 27
So, it must be the case that (x+y) = 9
If (x+y) = 9, then (x+y)² = 81
So, the answer to the target question is (x+y)² = 81
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

Posted from my mobile device

the solution is correct, however there is a calculation mistake where you subtract 9 from 27 rather than adding it.

Posted from my mobile device

Analyzing the question:
To figure out \((x + y)^2\) we only need \(x + y.\) Since we are given \(x^2 - y^2 = (x - y)(x + y) = 27\), knowing \(x - y\) would also give \(x + y\) from that equation. Therefore we are trying to find \(x - y\), \(x + y\), or both values of x and y.

Statement 1:
(1) This gives us \(x^2 = 36\) but we don’t know if x is positive or negative. The two solutions give different values of \((x + y)^2\) so this is insufficient.

Statement 2:
(2) As stated above this is sufficient.

Answer: B

Thanks for the heads up!
I have edited my response accordingly.

Cheers,
Brent

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