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

 It is currently 20 Aug 2019, 15:46 ### 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. ### Request Expert Reply # If x is an integer, how many possible values

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

### Hide Tags

Manager  G
Joined: 15 Dec 2015
Posts: 114
GMAT 1: 680 Q49 V34 GPA: 4
WE: Information Technology (Computer Software)
If x is an integer, how many possible values  [#permalink]

### Show Tags

4
27 00:00

Difficulty:   55% (hard)

Question Stats: 48% (01:06) correct 52% (01:15) wrong based on 474 sessions

### HideShow timer Statistics

If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

A. 4
B. 2
C. 3
D. 1
E. 0
##### Most Helpful Expert Reply
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3251
If x is an integer, how many possible values  [#permalink]

### Show Tags

5
1
DH99 wrote:
If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

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

One method: check the signs of the terms.

The squared term is positive (or nonnegative if x=0).

The term whose product is (+5) * (some nonnegative or positive number because absolute value nonnegative or positive), is positive or nonnegative (if x = 0).

The constant is positive.

You cannot sum three positive numbers, or two zeros (if x=0) and a positive, and get zero. No values will work.

Answer E

Another way: If "check the signs method" doesn't occur to you, try factoring the quadratic as if there were no absolute value bars around the x in the second term.

$$x^2+5x+6=0$$
(x + 3)(x + 2)

So x = -3 or -2

Check the values. Plug -3 and -2 into original equation.

Neither works:
-3: $$x^2+5|x|+6=0$$
-3: 9 + (5*3) + 6 =
-3: 9 + 15 + 6 does not equal zero.

From the pattern of positive terms that results when plugging in a negative number (squared term is positive, absolute value term is positive, constant is positive), -2 will not work either.

From the (+) + (+) + (+) pattern: you cannot get to 0 with three positive numbers. No values will work.

Answer E
_________________
SC Butler has resumed!
Get two SC questions to practice, whose links you can find by date, here.

What we do now echoes in eternity.—Marcus Aurelius

Originally posted by generis on 07 Aug 2017, 10:46.
Last edited by generis on 15 Sep 2018, 13:15, edited 3 times in total.
##### General Discussion
Manager  S
Joined: 30 Mar 2017
Posts: 66
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

2
E
When x <0
The equation will become x^2 -5x+6=0. The two roots are 3 and 2 going against the range of x .
If we take x>0, x^2 +5x+6=0 and two roots will be -3, and -2.
This too goes against the range of x. So zero solutions.
Another way of looking at this will be to observe that all the three terms of the equation are each greater than zero. So the equation will never be zero and hence no or zero solution.

Sent from my Moto G (5) Plus using GMAT Club Forum mobile app
Manager  S
Joined: 16 May 2017
Posts: 60
Location: India
GMAT 1: 710 Q47 V39 WE: General Management (Retail Banking)
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

By range of x what are you suggesting that? X is an integer.

Sent from my Redmi 4 using GMAT Club Forum mobile app
_________________
"The harder you work the luckier you get"
Manager  G
Joined: 15 Dec 2015
Posts: 114
GMAT 1: 680 Q49 V34 GPA: 4
WE: Information Technology (Computer Software)
If x is an integer, how many possible values  [#permalink]

### Show Tags

1
genxer123 wrote:
DH99 wrote:
If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

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

Looking at other posts, I might be oversimplifying here . . . Please correct me if I'm mistaken.

One method: check the signs of the terms.

The squared term is positive.

The term whose product is (+5) * (some positive number because absolute value is positive), is positive.

The constant is positive.

You cannot sum three positive numbers and get zero. No values will work.

Answer E

Another way: If "check the signs method" doesn't occur to you, try factoring the quadratic as if there were no absolute value bars around the x in the second term.

$$x^2+5x+6=0$$
(x + 3)(x + 2)

So x = -3 or -2

Check the values. Plug -3 and -2 into original equation.

Neither works:
-3: 9 + 15 + 6 does not equal zero.

From the pattern of positive terms that result when plugging in a negative number (squared term is positive, absolute value term is positive, constant is positive), -2 will not work either.

From the (+) + (+) + (+) pattern: you cannot get to 0 with three positive numbers. No values will work.

Answer E

genxer123
I like your "One method: check the signs of the terms." very much.+1 kudos given. So, it will be true for any quadratic equation in the form ax^2+b|x|+c=0 as long a,b and c are positive?
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3251
If x is an integer, how many possible values  [#permalink]

### Show Tags

1
DH99 wrote:
genxer123
I like your "One method: check the signs of the terms." very much.+1 kudos given. So, it will be true for any quadratic equation in the form ax^2+b|x|+c=0 as long a,b and c are positive?

Yes -- but also nonnegative (x = 0 especially, or a AND b = 0) . . . as long as c is positive.

In other words, if your first two terms result in the nonnegative 0, check c. Positive? No solution. You can't add a positive number to zero and get zero.

Sometimes you will see posters insist that there is no such thing as the absolute value of 0.

Because absolute value is a distance (from point of origin, often 0) there is such a thing: |0| is 0. Zero is 0 distance away from zero.

Come to think of it, though the coefficients wouldn't work in the factor method part, I'm going to amend the first part my answer to include nonnegative! Thanks. _________________
SC Butler has resumed!
Get two SC questions to practice, whose links you can find by date, here.

What we do now echoes in eternity.—Marcus Aurelius
Manager  B
Joined: 24 Jun 2017
Posts: 120
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

2
a tricky one x^2+5|x|+6=0

can be written as
(|x| + 3) (|x| + 2) = 0 as |x|^2 always = x^2
then no solution for 0 as there is no negative value for |x|
Senior Manager  P
Joined: 02 Apr 2014
Posts: 471
Location: India
Schools: XLRI"20
GMAT 1: 700 Q50 V34 GPA: 3.5
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

1
$$x^2 = |x||x$$|
given $$x^2 + 5|x| + 6 = 0$$

Method1:
Let a = |x|
a^2 + 5a + 6 = 0
roots: a = -2, -3
|x| = -2, |x| = -3
modulus can never be negative, so no solution exists

Method 2:
$$x^2 + 5|x| + 6 = 0$$ => to hold this, $$x^2 + 5|x| = -6$$, but modulus can never be negative as the minimum value is 0, so solution exists
Intern  B
Joined: 22 Mar 2017
Posts: 28
GMAT 1: 680 Q48 V35 Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

2
DHAR wrote:
If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

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

Humbly I think that it´s much easier to think in the way that $$x^2+5|x|+6=0$$ is gonna have to suffer a check for strenuous roots afterwards; thus, it´s not necessary to solve or to do absolutely anything with the equation, and it´s just enough that your bubble lights up with the strenuous nuance and say to youself:

"okay, the absolute-value term is always positive, and so is the quadratic term, so there is simply no way that a solution exists that can make the left side side equal to the right".

Conclusion: 0 solutions.

-
_________________
If it helped, some kudos would be more than welcome! King regards,

Rooigle
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3251
If x is an integer, how many possible values  [#permalink]

### Show Tags

1
RooIgle wrote:
DHAR wrote:
If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

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

Humbly I think that it´s much easier to think in the way that $$x^2+5|x|+6=0$$ is gonna have to suffer a check for strenuous roots afterwards; thus, it´s not necessary to solve or to do absolutely anything with the equation, and it´s just enough that your bubble lights up with the strenuous nuance and say to youself:

"okay, the absolute-value term is always positive, and so is the quadratic term, so there is simply no way that a solution exists that can make the left side side equal to the right".

Conclusion: 0 solutions.
-

RooIgle
Surely, a few travelers will linger, then halt, when a collection of oddly graceful words unfurls.
Surely, those stock-still few will wonder: of what stuff is this "strenuous nuance" made? Is it balletic, like a thought? Or sensible, like a nod?
Surely, as they wander away, they will dare to see beyond content (zero does equal zero, after all).
Surely, they will know enough to murmur, "Who gets that close to the oxymoronic but does not collide with it?"
Brave soul. Perilous territory.
No. Not "surely." But this time, yes. At least one. Nicely done.
_________________
SC Butler has resumed!
Get two SC questions to practice, whose links you can find by date, here.

What we do now echoes in eternity.—Marcus Aurelius
Senior Manager  S
Joined: 08 Jun 2015
Posts: 421
Location: India
GMAT 1: 640 Q48 V29 GMAT 2: 700 Q48 V38 GPA: 3.33
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

+1 for option E. The equation can be re-written as |x|^2+5|x|+6=0. Solving we get |x|=-3,-2. This clearly not possible. Hence no value of x will yield the required value. Hence option E.
_________________
" The few , the fearless "
CEO  V
Joined: 12 Sep 2015
Posts: 3912
Location: Canada
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

1
Top Contributor
DHAR wrote:
If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

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

Take: x² + 5|x| + 6 = 0
Subtract 6 from both sides to get: + 5|x| = -6

KEY CONCEPT: x² ≥ 0 and |x| ≥ 0 for all values of x
In other words, x² will always be greater than or equal to 0
And |x| will always be greater than or equal to 0, which means 5|x| will always be greater than or equal to 0

So, we can take our equation, + 5|x| = -6, and rewrite it as follows:
(some number that's greater than or equal to zero) + (some number that's greater than or equal to zero) = -6
As we can see, it's impossible for the left side of the equation to equal a NEGATIVE value.
As such, there can be no solution.

Answer: E

Cheers,
Brent
_________________
GMATH Teacher P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 937
Re: If x is an integer, how many possible values  [#permalink]

### Show Tags

DHAR wrote:
If x is an integer, how many possible values of x exist for $$x^2+5|x|+6=0 ?$$

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

$$?\,\,\,:\,\,\,\# \,\,\,{\mathop{\rm int}} \,\,\,{\rm{roots}}\,\,\,{\rm{for}}\,\,\,\,{x^2} + 5\left| x \right| + 6 = 0$$

$$\left. \matrix{ {x^2} \ge 0 \hfill \cr \left| x \right|\,\, \ge 0\,\, \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,?\,\,\,\,:\,\,\,{x^2} + 5\left| x \right|\, + 6\,\,\, \ge 6\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 0$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net Re: If x is an integer, how many possible values   [#permalink] 07 Oct 2018, 08:25
Display posts from previous: Sort by

# If x is an integer, how many possible values

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

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

#### MBA Resources  