Last visit was: 21 Apr 2026, 17:00 It is currently 21 Apr 2026, 17:00
Home
Messages
Tests
Schools
Events
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
User avatar
GK_Gmat
User avatar
Joined: 09 Aug 2006
Last visit: 28 Feb 2009
Posts: 348
Own Kudos:
1,078
 [143]
Posts: 348
Kudos: 1,078
 [143]
Which of the following is always equal to sqrt(9+x^2-6x)? Updated on: 14 May 2023, 12:39
Originally posted by GK_Gmat on 22 Oct 2007, 23:02.
Last edited by Bunuel on 14 May 2023, 12:39, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
11
Kudos
Add Kudos
132
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

35% (medium)

Question Stats:

63% (01:01) correct 37% (01:04) wrong based on 2768 sessions

History

Date
Time
Result
Not Attempted Yet
Which of the following options always equates to \(\sqrt{9 + x^2 - 6x}\)?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x

Experience GMAT Club Test Questions
Yes, you've landed on a GMAT Club Tests question
Craving more? Unlock our full suite of GMAT Club Tests here
Want to experience more? Get a taste of our tests with our free trial today
Rise to the challenge with GMAT Club Tests. Happy practicing!
GMAT Club Tests Check Our Products Try Free Tests
ShowHide Answer
Official Answer
C
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 21 Apr 2026
Posts: 109,728
Own Kudos:
810,467
 [16]
Given Kudos: 105,800
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,728
Kudos: 810,467
 [16]
Which of the following is always equal to sqrt(9+x^2-6x)? 19 Feb 2013, 05:09
6
Kudos
Add Kudos
10
Bookmarks
Bookmark this Post
Sachin9
This is true and absolute value properties confirms this because:
|a-b| = |b-a|

Bunuel/KArishma,
Is this always true?
Show more

Yes, since both |a-b| and |b-a| represent the distance between a and b on the number line.

COMPLETE SOLUTION:

Which of the following options always equates to \(\sqrt{9 + x^2 - 6x}\)?

A. \(x - 3\)
B. \(3 + x\)
C. \(|3 - x|\)
D. \(|3 + x|\)
E. \(3 - x\)


\(\sqrt{9 + x^2 - 6x} = \)

\(=\sqrt{(3 - x)^2} = \)

\(=|3 - x|\)

Note that both \(|x - 3|\) and \(|3 - x|\) represent the distance between \(x\) and 3. Hence, it follows that \(|x - 3| = |3 - x|\).


Answer: C
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 21 Apr 2026
Posts: 109,728
Own Kudos:
810,467
 [10]
Given Kudos: 105,800
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,728
Kudos: 810,467
 [10]
Which of the following is always equal to sqrt(9+x^2-6x)? 07 Apr 2015, 10:48
5
Kudos
Add Kudos
5
Bookmarks
Bookmark this Post
ankittiss
Bunuel Can you please help here?
According to my understanding there are two cases in the GMAT:
Case 1 :

sqrt[(-5)*(-5)] = -5

Case 2 :

sqrt[(5)*(5)] = 5

Is my understanding correct?

Thanks in advance
Ankit
Show more

No, it's totally wrong.

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 positive root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

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 \(x=\sqrt{9}=3\).

Theory on Number Properties: math-number-theory-88376.html
Tips on Numper Properties: number-properties-tips-and-hints-174996.html

All DS Number Properties Problems to practice: search.php?search_id=tag&tag_id=38
All PS Number Properties Problems to practice: search.php?search_id=tag&tag_id=59

About \(\sqrt{x^2}=|x|\):

Again, the point here is that since square root function can not give negative result then \(\sqrt{some \ expression}\geq{0}\).

So \(\sqrt{x^2}\geq{0}\). But what does \(\sqrt{x^2}\) equal to?

Let's consider following examples:
If \(x=5\) --> \(\sqrt{x^2}=\sqrt{25}=5=x=positive\);
If \(x=-5\) --> \(\sqrt{x^2}=\sqrt{25}=5=-x=positive\).

So we got that:
\(\sqrt{x^2}=x\), if \(x\geq{0}\);
\(\sqrt{x^2}=-x\), if \(x<0\).

What function does exactly the same thing? The absolute value function: \(|x|=x\), if \(x\geq{0}\) and \(|x|=-x\), if \(x<0\). That is why \(\sqrt{x^2}=|x|\).

Hope it helps.
General Discussion
User avatar
bkk145
User avatar
Joined: 10 Jun 2007
Last visit: 23 Feb 2014
Posts: 645
Own Kudos:
1,801
 [27]
Posts: 645
Kudos: 1,801
 [27]
Which of the following is always equal to sqrt(9+x^2-6x)? 23 Oct 2007, 11:53
19
Kudos
Add Kudos
8
Bookmarks
Bookmark this Post
The answer is indeed
|x-3|

Don't for get that...

|x-3| = |3-x|

C is the answer.
User avatar
Sachin9
User avatar
Joined: 22 Jul 2012
Last visit: 25 Dec 2015
Posts: 349
Own Kudos:
175
 [3]
Given Kudos: 562
Status:Gonna rock this time!!!
Location: India
GMAT 1: 640 Q43 V34
GMAT 2: 630 Q47 V29
WE:Information Technology (Computer Software)
GMAT 2: 630 Q47 V29
Posts: 349
Kudos: 175
 [3]
Which of the following is always equal to sqrt(9+x^2-6x)? 19 Feb 2013, 03:04
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
This is true and absolute value properties confirms this because:
|a-b| = |b-a|

Bunuel/KArishma,
Is this always true?
avatar
mehasingh
avatar
Joined: 16 Dec 2012
Last visit: 22 Feb 2013
Posts: 3
Given Kudos: 1
Posts: 3
Kudos: 0
Which of the following is always equal to sqrt(9+x^2-6x)? 20 Feb 2013, 21:31
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Which of the following is always equal to \sqrt{9+x^2-6x}?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x


Hi, can anyone explain me how to go abt this...the above answers have got a
lil confusing for me.

ill go one option at a time:

A. x-3

when we square x-3 it give x^2+9-6x
if we squareroot x^2+9-6x then we get the same exp...so for me this seems
to be an answer

B. 3+x

when we square 3+x it gives 9+6x+x^2
this is all positive unlike 9+x^2-6x given in the question, so not an answer

C. |3-x|

given sqrt{9+x^2-6x}

solving sqrt(3-x)

|3-x|=sqrt(3-x)

this too seems to be a possible answer

D. |3+x|

this will give 9+x^2+6x which is not equal to the equation given hence not
an answer

E. 3-x

=9+x^2-6x

this too seems fine

can anyone please clarify where im going wrong.... in all the options...


Thanks
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 21 Apr 2026
Posts: 109,728
Own Kudos:
810,467
 [3]
Given Kudos: 105,800
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,728
Kudos: 810,467
 [3]
Which of the following is always equal to sqrt(9+x^2-6x)? 21 Feb 2013, 03:28
1
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
mehasingh
Which of the following is always equal to \sqrt{9+x^2-6x}?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x


Hi, can anyone explain me how to go abt this...the above answers have got a
lil confusing for me.

ill go one option at a time:

A. x-3
Show more

Notice that the square root function cannot give negative result: \(\sqrt{{some \ expression}}\geq{0}\).

So, \(\sqrt{9+x^2-6x}=\sqrt{(3-x)^2}\geq{0}\).

Now, in option A we have x-3, which can be negative if x<3, so A cannot be the correct answer.

Hope it's clear.
avatar
ankittiss
avatar
Joined: 19 Jul 2014
Last visit: 12 May 2020
Posts: 21
Own Kudos:
50
Given Kudos: 138
Location: India
Concentration: Finance, Accounting
Schools: Mays '19 Duke '19 Ross '19
GPA: 3.12
Schools: Mays '19 Duke '19 Ross '19
Posts: 21
Kudos: 50
Which of the following is always equal to sqrt(9+x^2-6x)? 07 Apr 2015, 10:36
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel Can you please help here?
According to my understanding there are two cases in the GMAT:
Case 1 :

sqrt[(-5)*(-5)] = -5

Case 2 :

sqrt[(5)*(5)] = 5

Is my understanding correct?

Thanks in advance
Ankit
User avatar
LogicGuru1
User avatar
Joined: 04 Jun 2016
Last visit: 28 May 2024
Posts: 463
Own Kudos:
2,643
 [1]
Given Kudos: 36
GMAT 1: 750 Q49 V43
GMAT 1: 750 Q49 V43
Posts: 463
Kudos: 2,643
 [1]
Which of the following is always equal to sqrt(9+x^2-6x)? 25 Jul 2016, 02:39
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GK_Gmat
Which of the following is always equal to \(\sqrt{9+x^2-6x}\)?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x
Show more

There should be no confusion regarding this question.
SQUAREROOT and MODULUS have the same property regarding the polarity of a number. THEY BOTH YIELDS ONLY POSITIVE OUTPUTS.

Q:- what is the surest way to make any value positive. ?
A:- Take the modulus of the value |x|

So we know \(\sqrt{9+x^2-6x}\) will gives us ONLY POSITIVE VALUE, Then we should make sure that the option also matches this property.
Therefore only |3 - x| and |3+x| are the one that will always give positive value
BUT |3+x| is not a root or solution of \(\sqrt{9+x^2-6x}\)
So the only option left is |3-x|

ANSWER IS C
User avatar
sixfivefive
User avatar
Joined: 01 May 2025
Last visit: 26 Jan 2026
Posts: 8
Own Kudos:
3
Given Kudos: 2
Posts: 8
Kudos: 3
Which of the following is always equal to sqrt(9+x^2-6x)? 05 May 2025, 07:29
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Isn't it ok if (x - 3) is negative though because you are squaring it prior to taking the square root? It will never be the square root of a negative.
Bunuel
mehasingh
Which of the following is always equal to \sqrt{9+x^2-6x}?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x


Hi, can anyone explain me how to go abt this...the above answers have got a
lil confusing for me.

ill go one option at a time:

A. x-3

Notice that the square root function cannot give negative result: \(\sqrt{{some \ expression}}\geq{0}\).

So, \(\sqrt{9+x^2-6x}=\sqrt{(3-x)^2}\geq{0}\).

Now, in option A we have x-3, which can be negative if x<3, so A cannot be the correct answer.

Hope it's clear.
Show more
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 21 Apr 2026
Posts: 109,728
Own Kudos:
810,467
Given Kudos: 105,800
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,728
Kudos: 810,467
Which of the following is always equal to sqrt(9+x^2-6x)? 05 May 2025, 07:52
Kudos
Add Kudos
Bookmarks
Bookmark this Post
sixfivefive
Isn't it ok if (x - 3) is negative though because you are squaring it prior to taking the square root? It will never be the square root of a negative.
Bunuel
mehasingh
Which of the following is always equal to \sqrt{9+x^2-6x}?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x


Hi, can anyone explain me how to go abt this...the above answers have got a
lil confusing for me.

ill go one option at a time:

A. x-3

Notice that the square root function cannot give negative result: \(\sqrt{{some \ expression}}\geq{0}\).

So, \(\sqrt{9+x^2-6x}=\sqrt{(3-x)^2}\geq{0}\).

Now, in option A we have x-3, which can be negative if x<3, so A cannot be the correct answer.

Hope it's clear.
Show more

\(\sqrt{9 + x^2 - 6x} = \)

\(=\sqrt{(3 - x)^2} = \)

\(=|3 - x|=|x-3|\)

If x - 3 is negative then |x - 3| = -(x - 3) = 3 - x, which in this case will be positive. So, in any case we get posiitve result.
User avatar
luisdicampo
User avatar
Joined: 10 Feb 2025
Last visit: 19 Apr 2026
Posts: 480
Own Kudos:
73
Given Kudos: 328
Products:
My Reviews Forum Quiz
Posts: 480
Kudos: 73
Which of the following is always equal to sqrt(9+x^2-6x)? 21 Nov 2025, 03:33
Kudos
Add Kudos
Bookmarks
Bookmark this Post
GK_Gmat
Which of the following options always equates to \(\sqrt{9 + x^2 - 6x}\)?

A. x - 3
B. 3 + x
C. |3 - x|
D. |3 + x|
E. 3 - x

Experience GMAT Club Test Questions
Yes, you've landed on a GMAT Club Tests question
Craving more? Unlock our full suite of GMAT Club Tests here
Want to experience more? Get a taste of our tests with our free trial today
Rise to the challenge with GMAT Club Tests. Happy practicing!
GMAT Club Tests Check Our Products Try Free Tests
Show more
Method 1: Algebraic Approach

The expression inside the square root is a perfect square trinomial.
Rearrange the terms:
\(\sqrt{9 + x^2 - 6x} = \sqrt{x^2 - 6x + 9}\)

Factor the quadratic expression:
\(x^2 - 6x + 9 = (x - 3)^2\) (or equivalently \((3 - x)^2\))

So the expression becomes:
\(\sqrt{(x - 3)^2}\)

CRITICAL RULE:
In GMAT math, \(\sqrt{u^2}\) is defined as \(|u|\), not just \(u\). The square root symbol \(\sqrt{}\) only yields the non-negative root.

Therefore:
\(\sqrt{(x - 3)^2} = |x - 3|\)

Note that \(|x - 3|\) is mathematically identical to \(|3 - x|\).
If you check the options, \(|3 - x|\) is Option (C).

[hr]

Method 2: Picking Numbers
If you forget the absolute value rule, picking numbers will save you.

Case 1: Pick \(x = 0\) (Choose a value smaller than 3)
Substitute into the original expression:
\(\sqrt{9 + 0^2 - 6(0)} = \sqrt{9} = 3\)

Now check the options:
A. \(0 - 3 = -3\) (Incorrect)
B. \(3 + 0 = 3\) (Keep)
C. \(|3 - 0| = 3\) (Keep)
D. \(|3 + 0| = 3\) (Keep)
E. \(3 - 0 = 3\) (Keep)

Case 2: Pick \(x = 5\) (Choose a value larger than 3)
Substitute into the original expression:
\(\sqrt{9 + 25 - 30} = \sqrt{4} = 2\)

Now check the remaining options:
B. \(3 + 5 = 8\) (Incorrect)
C. \(|3 - 5| = |-2| = 2\) (Correct)
D. \(|3 + 5| = 8\) (Incorrect)
E. \(3 - 5 = -2\) (Incorrect, remember the result of a square root must be positive)

Answer: C
Moderators:
Bunuel
Math Expert
109728 posts
Krunaal
Tuck School Moderator
853 posts