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# M19-20

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Math Expert
Joined: 02 Sep 2009
Posts: 42269

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16 Sep 2014, 01:06
Expert's post
12
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Difficulty:

45% (medium)

Question Stats:

59% (00:30) correct 41% (00:35) wrong based on 196 sessions

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Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?

A. $$x - 3$$
B. $$3 + x$$
C. $$|3 - x|$$
D. $$|3 + x|$$
E. $$3 - x$$
[Reveal] Spoiler: OA

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Kudos [?]: 132827 [0], given: 12378

Math Expert
Joined: 02 Sep 2009
Posts: 42269

Kudos [?]: 132827 [0], given: 12378

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16 Sep 2014, 01:06
Expert's post
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Official Solution:

Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?

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

$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).

_________________

Kudos [?]: 132827 [0], given: 12378

Current Student
Joined: 04 Jul 2014
Posts: 294

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Location: India
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GMAT 2: 640 Q44 V34
GMAT 3: 710 Q49 V37
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WE: Analyst (Accounting)

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21 Nov 2014, 03:18
1
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Hi Bunuel,

I got the roots for the equation as (x-3)^2 (reading the equation as x^2 - 6x +9 - We've flipped the values of a and b in the equation a^2 - 2ab + b^2). Based on my approach |x - 3| would be the correct answer and on yours |3 - x| is the correct answer.

Does my above para make any sense? If it does, and if the question has both of these options, which would be the correct answer? If it doesn't please help me understand my mistake.
_________________

Cheers!!

JA
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Math Expert
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21 Nov 2014, 04:51
3
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joseph0alexander wrote:
Hi Bunuel,

I got the roots for the equation as (x-3)^2 (reading the equation as x^2 - 6x +9 - We've flipped the values of a and b in the equation a^2 - 2ab + b^2). Based on my approach |x - 3| would be the correct answer and on yours |3 - x| is the correct answer.

Does my above para make any sense? If it does, and if the question has both of these options, which would be the correct answer? If it doesn't please help me understand my mistake.

The point is that |x - 3| = |3 - x|. Both indicate the distance between x and 3.
_________________

Kudos [?]: 132827 [3], given: 12378

Current Student
Joined: 04 Jul 2014
Posts: 294

Kudos [?]: 345 [1], given: 413

Location: India
GMAT 1: 640 Q47 V31
GMAT 2: 640 Q44 V34
GMAT 3: 710 Q49 V37
GPA: 3.58
WE: Analyst (Accounting)

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21 Nov 2014, 05:05
1
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Bunuel wrote:
The point is that |x - 3| = |3 - x|. Both indicate the distance between x and 3.

So, I understand that both of our answers are correct and that both these values won't appear as options in a single question.
_________________

Cheers!!

JA
If you like my post, let me know. Give me a kudos!

Kudos [?]: 345 [1], given: 413

Math Expert
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21 Nov 2014, 05:07
1
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Expert's post
joseph0alexander wrote:
Bunuel wrote:
The point is that |x - 3| = |3 - x|. Both indicate the distance between x and 3.

So, I understand that both of our answers are correct and that both these values won't appear as options in a single question.

____________
Yes, that's correct.
_________________

Kudos [?]: 132827 [1], given: 12378

Current Student
Joined: 14 May 2014
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24 Jul 2015, 07:44
Bunuel wrote:
Official Solution:

Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?

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

$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).

i answered choice a : x-3

i took the the expression as x^2-6x+9 ---> (x-3)^2. no my understanding is that on GMAT when some expression is under root sign, only positive root is considered...? is it correct. if it is so, the we need not take mod value...? pls help

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Math Expert
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Kudos [?]: 132827 [0], given: 12378

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24 Jul 2015, 07:51
Expert's post
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riyazgilani wrote:
Bunuel wrote:
Official Solution:

Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?

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

$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).

i answered choice a : x-3

i took the the expression as x^2-6x+9 ---> (x-3)^2. no my understanding is that on GMAT when some expression is under root sign, only positive root is considered...? is it correct. if it is so, the we need not take mod value...? pls help

Exactly because the square root function cannot give negative result the answer is |3-x|, an absolute value of 3-x, which also cannot be negative.

MUST KNOW: $$\sqrt{x^2}=|x|$$:

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 $$\sqrt{x^2}=|x|$$.

I'd advice to go through basics and only after to practice questions:

Theory on Abolute Values: math-absolute-value-modulus-86462.html
Absolute value tips: absolute-value-tips-and-hints-175002.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

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Intern
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10 Dec 2015, 03:55
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. please elaborate the explanation.

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Math Expert
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Kudos [?]: 132827 [0], given: 12378

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12 Dec 2015, 08:37
Bhavanasg wrote:
I think this is a high-quality question and the explanation isn't clear enough, please elaborate. please elaborate the explanation.

Hope it helps.
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22 Aug 2016, 14:02
I think this is a high-quality question and I agree with explanation.

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Intern
Joined: 18 Jun 2015
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12 Sep 2016, 15:23
1
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After solving this question I got the result as |X-3| and I answered it as X-3, And got this wrong.
Thanks for giving the link which clears the basic of Absolute value including the result:

|A-B| = |B-A|

Which is clearly applicable in this question provided options.

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Manager
Joined: 14 Oct 2012
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30 Mar 2017, 20:19
1
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Bunuel wrote:
riyazgilani wrote:
Bunuel wrote:
Official Solution:

Which of the following always equals $$\sqrt{9 + x^2 - 6x}$$ ?

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

$$\sqrt{9 + x^2 - 6x} = \sqrt{(3 - x)^2} = |3 - x|$$ (by the definition of a square root).

i answered choice a : x-3

i took the the expression as x^2-6x+9 ---> (x-3)^2. no my understanding is that on GMAT when some expression is under root sign, only positive root is considered...? is it correct. if it is so, the we need not take mod value...? pls help

Exactly because the square root function cannot give negative result the answer is |3-x|, an absolute value of 3-x, which also cannot be negative.

MUST KNOW: $$\sqrt{x^2}=|x|$$:

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 $$\sqrt{x^2}=|x|$$.

I'd advice to go through basics and only after to practice questions:

Hello Bunuel, Vyshak
I thought the equation was as follows:
$$\sqrt{x^2}=x$$, if $$x>{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<={0}$$.
Am i wrong? Please correct me if so.
Thanks

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Math Expert
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Posts: 42269

Kudos [?]: 132827 [0], given: 12378

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30 Mar 2017, 22:01
manishtank1988 wrote:
Hello Bunuel, Vyshak
I thought the equation was as follows:
$$\sqrt{x^2}=x$$, if $$x>{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<={0}$$.
Am i wrong? Please correct me if so.
Thanks

You can put = sign in any of the two because $$\sqrt{0^2}=0=-0$$.
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31 Mar 2017, 10:00
Dear Bunuel,

The answer choices CAN"T have the following answers in their list: $$|3 - x|$$ and $$|x - 3|$$ Because the following rule:

$$|3 - x|$$ = $$|x - 3|$$

Am I right?

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31 Mar 2017, 15:07
Bunuel wrote:
manishtank1988 wrote:
Hello Bunuel, Vyshak
I thought the equation was as follows:
$$\sqrt{x^2}=x$$, if $$x>{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<={0}$$.
Am i wrong? Please correct me if so.
Thanks

You can put = sign in any of the two because $$\sqrt{0^2}=0=-0$$.

Understood thanks...

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21 Aug 2017, 05:28
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This question is based on the principle of $$\sqrt{x^2}=|x|$$
Hence its $$|x-3|=|3-x|$$
Straight away C.

Press Kudos id this helps!

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Re: M19-20   [#permalink] 21 Aug 2017, 05:28
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