GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 17 Oct 2019, 06:47

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

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

What is the range of all the roots of |x^2 - 2| = x ?

Author Message
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 58414
What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

21 May 2015, 05:15
5
34
00:00

Difficulty:

95% (hard)

Question Stats:

37% (01:54) correct 63% (01:43) wrong based on 693 sessions

HideShow timer Statistics

What is the range of all the roots of |x^2 - 2| = x ?

A. 4
B. 3
C. 2
D. 1
E. 0

_________________
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15263
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

Updated on: 25 May 2015, 10:53
7
4
Hi All,

You have to be VERY careful with this question. It's actually built more to test your attention-to-detail than your "math skills"....

We're asked to find the RANGE of the roots of the following equation: |x^2 - 2| = x ?

Before you jump in and start doing calculations, there are a couple of points to note:

1) Absolute value calculations can NEVER equal a negative number. Here, we have an EQUATION set equal to X. As such, X CANNOT be NEGATIVE.
2) The answer choices are small integers, so the roots of the above equation are likely also integers that are relatively close to one another.
3) Since the question mentions ROOTS, there should be at least 2 solutions.

A bit of basic "brute force" is all that's really needed to find the roots of the equation....

Could X = 0?
|0-2| is NOT 0, so X cannot be 0

Could X = 1?
|1-2| does = 1, so X = 1 is a root

Could X = 2?
|4-2| does = 2, so X = 2 is a root

Could X = 3?
|9-2| is NOT 3, so X cannot be 3

As X gets bigger, the absolute value calculation gets even larger (and farther 'away') from X.

Thus, the only roots are 1 and 2. The range is 2-1 = 1

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com

The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★

Originally posted by EMPOWERgmatRichC on 22 May 2015, 11:54.
Last edited by EMPOWERgmatRichC on 25 May 2015, 10:53, edited 1 time in total.
Manager
Joined: 18 Mar 2014
Posts: 226
Location: India
Concentration: Operations, Strategy
GMAT 1: 670 Q48 V35
GPA: 3.19
WE: Information Technology (Computer Software)
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

21 May 2015, 06:55
6
3
we get 2 quadratic equations here ..
1) x^2-x-2=0 ....... roots 2 , -1
2) x^2+x-2=0 ........ roots -2, 1

Inserting each root in given equation , it can be seen that -1 and -2 do not satisfy the equations .
So value of x for given equation .... x=2 or x=1
I guess range is 2-1 =1
_________________
Press +1 Kudos if you find this Post helpful
General Discussion
Current Student
Joined: 13 Nov 2014
Posts: 108
GMAT 1: 740 Q50 V40
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

21 May 2015, 06:27
Bunuel wrote:
What is the range of all the roots of |x^2 - 2| = x ?

A. 4
B. 3
C. 2
D. 1
E. 0

In short, we will be solving for 2 quadratics. The two are:

1) x^2-x-2=0
2) x^2+x-2=0

The roots for quadratic 1 are {-2,1} and the roots for quadratic 2 are {2,-1}.

Range is highest - lowest, i.e. 2-(-2) =4

answer is A... or so I hope
_________________
Gmat prep 1 600
Veritas 1 650
Veritas 2 680
Gmat prep 2 690 (48Q 37V)
Gmat prep 5 730 (47Q 42V)
Gmat prep 6 720 (48Q 41V)
Current Student
Joined: 13 Nov 2014
Posts: 108
GMAT 1: 740 Q50 V40
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

21 May 2015, 06:57
we get 2 quadratic equations here ..
1) x^2-x-2=0 ....... roots 2 , -1
2) x^2+x-2=0 ........ roots -2, 1

Inserting each root in given equation , it can be seen that -1 and -2 do not satisfy the equations .
So value of x for given equation .... x=2 or x=1
I guess range is 2-1 =1

Yeah you're right. Totally forgot to go back to the stem. -1 and -2 don't satisfy.
_________________
Gmat prep 1 600
Veritas 1 650
Veritas 2 680
Gmat prep 2 690 (48Q 37V)
Gmat prep 5 730 (47Q 42V)
Gmat prep 6 720 (48Q 41V)
Director
Joined: 07 Aug 2011
Posts: 502
GMAT 1: 630 Q49 V27
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

21 May 2015, 11:56
2
Attached image.
there are only 2 roots.
We can solve it algebrically as well.

Attachments

Screenshots_2015-05-22-02-42-21.png [ 231.56 KiB | Viewed 5638 times ]

Intern
Joined: 19 Mar 2015
Posts: 13
Location: United States
Concentration: Sustainability, Sustainability
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

22 May 2015, 09:56
1
we get 2 quadratic equations here ..
1) x^2-x-2=0 ....... roots 2 , -1
2) x^2+x-2=0 ........ roots -2, 1

Inserting each root in given equation , it can be seen that -1 and -2 do not satisfy the equations .
So value of x for given equation .... x=2 or x=1
I guess range is 2-1 =1

I guess all the four integers, -2,-1,1 & 2 satisfies the equation mod (x^2-2) = X. Please let me know if I am missing some point. Though I understood the explanation by graph method, I am not able to understand the way you have explained it.
Math Expert
Joined: 02 Sep 2009
Posts: 58414
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

25 May 2015, 07:29
5
3
Bunuel wrote:
What is the range of all the roots of |x^2 - 2| = x ?

A. 4
B. 3
C. 2
D. 1
E. 0

OFFICIAL SOLUTION:

First of all notice that since x is equal to an absolute value of some number (|x^2 - 2|), then x cannot be negative.

Next, |x^2 - 2| = x means that either x^2 - 2 = x or -(x^2 - 2) = x.

First equation gives x = -1 or x = 2. Since x cannot be negative, we are left with only x = 2.
Second equation gives x = -2 or x = 1. Again, since x cannot be negative, we are left with only x = 1.

The range = {largest} - {smallest} = 2 - 1 = 1.

_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 13241
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

30 Jan 2019, 23:23
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: What is the range of all the roots of |x^2 - 2| = x ?   [#permalink] 30 Jan 2019, 23:23
Display posts from previous: Sort by