What is the value of x^2 − y^2 ? (1) (x - y)^2 = 9 (2) x + y = 6

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

Statement 1 :

\((x-y)^2 = 9\)

x - y = +3 or -3

At the same time we don't know about x-y

NOT sufficient.

Statement 2 :

x + y = 6.

No information regarding x - y .Not sufficient.

1+ 2:

As we have 2 values for x - y , it's not possible to find out any specific value of \(x^2 - y^2\)

The best answer is E.

Please can someone explain to me why statement 1 is not sufficient.

The question stem is "What is the value of x^2− y^2 "

1) (X-Y)^2 = 9

If multiplied out gives us:
(X-Y)(X+Y) = 9

Multiply the brackets

X^2 +XY - XY -Y^2 = 9

X^2 - Y^2 = 9

Therefore X^2 - Y^2 = 9.

The question asks what is X^2 - Y^2? Is it not 9 as per statement 1, therefore sufficient?

Hi,

Why cant this question be solved by substituting statement 2
x=6-y
In statement 1 as,
x^2 + y^2 -2xy =9
(6-y)^2 + y^2 =9

We can get the value for x and y, right?

There would be an irrational value for x and y.

x=(-3(3)^1/2))/2 y=(12+(3(3)^1/2))/2

Then wont the data be sufficient with the 2 statements?
x^2-y^2=-198/4=49.5
Can someone please help me understand?

What is the value of \(x^2 - y^2\) ?
What is the value of (x+y)(x-y)?

(1) \((x - y)^2 = 9\)
(x - y) = +3 or -3, INSUFFICIENT

(2) x + y = 6
We don't know what (x-y) is; INSUFFICIENT.

(1&2) We know (x-y) = +3, -3
(x+y) = 6

We have 18, -18. INSUFFICIENT.

Answer is E.

Solution:

We need to determine the value of x^2 - y^2.

Statement One Alone:

We see that if x = 3 and y = 0, then x^2 - y^2 = 9 - 0 = 9. However, if x = 4 and y = 1, then x^2 - y^2 = 16 - 1 = 15. Statement one alone is not sufficient.

Statement Two Alone:

We see that if x = 6 and y = 0, then x^2 - y^2 = 36 - 0 = 26. However, if x = 4 and y = 2, then x^2 - y^2 = 16 - 4 = 12. Statement two alone is not sufficient.

Statements One and Two Together:

Notice that x^2 - y^2 = (x + y)(x - y). We are given that x + y = 6 in statement two, so we have x^2 - y^2 = 6(x - y). However, if we take the square root of the equation in statement one, we have:

x - y = ±3

If x - y = 3, then x^2 - y^2 = 6(3) = 18. However, if x - y = -3 , then x^2 - y^2 = 6(-3) = -18. Both statements together are still not sufficient.

Answer: E

As per basic Algebra, \(x^2 – y^2\)= (x-y) (x+y). Any information that provides us the values of these two expressions will be sufficient.

From statement I alone, \((x-y)^2\) = 9. Therefore, (x-y) = ± 3. No information about (x+y).
Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, (x+y) = 6. No information about (x-y).
Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:

From statement II, we have (x+y) = 6
From statement I, we have (x-y) = 3 or – 3.
No unique value for (x-y) (x+y). The combination of statements is insufficient. Answer option E can be eliminated.

The correct answer option is E.

Hope that helps!
Aravind BT

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