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

 It is currently 19 Oct 2019, 05:43 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  What is the range of all the roots of |x^2 - 2| = x ?

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 58449
What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

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 V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15281
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

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
_________________

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

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 S
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

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 S
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

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
Concentration: International Business, Technology
GMAT 1: 630 Q49 V27 Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

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 5647 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

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 V
Joined: 02 Sep 2009
Posts: 58449
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

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: 13271
Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

Show Tags

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

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne  