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

Question

What is the value of  x^2 - y^2  ?

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

(2) x + y = 6

(DS03308)

chetan2u · Accepted Answer

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

STATEMENTS..

(1)  (x - y)^2 = 9 
x-y= 3..
x-y=-3..
 various combinations of x and y possible fitting in any of two above 
insuff

(2) x + y = 6
 we do not know what is x-y 
insuff

combined..
we know x+y=6..
 but x-y can be 3, then ans is 18
OR x-y can be -3 ans will be -18 
insuff

E

thangvietnam · Answer

lets consider both case 1 and 2

x=6-y

(6-y-y)^2=9
6-2y=3 or -3
we can have two values of y and x

so, not sufficient

e

KSBGC · Answer

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.

pepsi123 · Answer

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?

vandanarjn92 · Answer

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?

Basshead · Answer

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.

ScottTargetTestPrep · Answer

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

CrackverbalGMAT · Answer

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

KanikaJain · Answer

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

(x+y)(x-y) = ?

Statement 1:
x - y = 3, -3

Statement 2:
x + y = 6

Statement 1 + 2

3*6, -3*6 thus 18, -18

Thus E

sherlocked221B · Answer

Bunuel KarishmaB bb

https://gmatclub.com/forum/square-root- ... ml#p924906

"According to this post and several other reliable sources — including comments by Bunuel on a few posts I have seen earlier — a square root should be considered non-negative on the GMAT. Given that, shouldn't the answer to the question be Option C?

It's still a bit confusing to determine exactly when this principle applies and when it doesn't."

Appreciate your help! Thank you!

bb · Answer

So the square root is not really relevant in the quadratic equation because you find the roots of the quadratic equation not by putting both sides under a square root, but by simplifying or finding the roots through the formula as otherwise you will end up missing roots.

Here is what Google Ai thinks about it:

When you can use the square root method with quadratic equations:

If the quadratic equation can be simplified to the form x^2 = k, where 'k' is a constant, you can take the square root of both sides to isolate 'x'.

For example, if you have x^2 = 9, you can take the square root of both sides to get x = ±3.

When you cannot use the square root method directly:

If the quadratic equation has a linear term (bx), like x^2 + bx + c = 0, you cannot directly take the square root of both sides.

In these cases, you need to use other methods like factoring, completing the square, or the quadratic formula.

Why you can't directly use the square root method in all cases:

The square root method is only useful when the squared term is isolated on one side of the equation and the other side is a constant. 
If there are other terms with the variable 'x', they need to be removed or manipulated to get the equation into the correct form. 
For example, if you have x^2 + 2x + 1 = 0, you can't directly take the square root of both sides. You would need to factor the left side to get (x + 1)^2 = 0, and then take the square root.

KarishmaB · Answer

Check out this post too: https://anaprep.com/algebra-squares-and-square-roots/

Bunuel · Answer

You are missing crucial point from the link you give:

x^2 = 16  has two solutions: x = 4 or -4

x = \sqrt{16}  has only one solution: x = 4

Here is a long answer. Study it carefully.

Even roots cannot give negative result.

\sqrt{...}  is the square root sign, a function (called the principal square root function), which cannot give negative result. So,  this sign ( \sqrt{...} ) always means non-negative square root .

https://gmatclub.com/forum/download/file.php?id=39502 
 The graph of the function f(x) = √x

Notice that it’s defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root.   That is:

\sqrt{9} = 3 , NOT +3 or -3;
 \sqrt {16} = 2 , NOT +2 or -2;

Similarly  \sqrt{\frac{1}{16}} = \frac{1}{4} , NOT +1/4 or -1/4.

Notice that in contrast, the equation  x^2 = 9  has TWO solutions, +3 and -3. Because  x^2 = 9  means that  x =-\sqrt{9}=-3  or  x=\sqrt{9}=3 .

Let's check the original question:

What is the value of x^2 − y^2 ?

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

This imply that x - y = 3 or x - y = -3. Not sufficient to get the value of x^2 - y^2.

(2) x + y = 6

This is also clearly insufficient.

(1)+(2) If x - y = 3, then (x - y)(x + y) = x^2 - y^2 = 18. However, if x - y = -3, then (x - y)(x + y) = x^2 - y^2 = -18. Not sufficient.

Answer: E.

And finally, pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.

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

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GMAT Data Sufficiency (DS) Questions

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

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