GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 29 Jan 2020, 04:35

### 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 sum of all solutions to the equation x^(2x² + 4x – 6) = x

Author Message
TAGS:

### Hide Tags

GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4235
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

Updated on: 13 Nov 2019, 16:57
3
1
Top Contributor
20
00:00

Difficulty:

95% (hard)

Question Stats:

20% (02:14) correct 80% (02:08) wrong based on 219 sessions

### HideShow timer Statistics

What is the sum of all solutions to the equation $$x^{(2x^2 + 4x – 6)} = x^{(x^2 + 8x +6)}$$ ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

_________________
Test confidently with gmatprepnow.com

Originally posted by GMATPrepNow on 02 Jan 2017, 13:45.
Last edited by GMATPrepNow on 13 Nov 2019, 16:57, edited 2 times in total.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4235
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

03 Jan 2017, 11:20
3
Top Contributor
5
GMATPrepNow wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

IMPORTANT: If b^x = b^y, then x = y, as long as b ≠ 0, b ≠ 1 and b ≠ -1
For example, if we have 1^x = 1^y, we cannot conclude that x = y, since 1^x equals 1^y FOR ALL values of x and y. For example, 1² = 1³, but we can't conclude that 2 = 3.

So, let's first see what happens when the base (x) equals 0, 1 and -1

If x = 0, then we have: 0^(2(0²) + 4(0) – 6) = 0^(0² + 8(0) + 6)
Simplify: 0^(-6) = 0^6
Evaluate: 0 = 0
So, x = 0 is one solution to the equation (yes, I know that x = 0 does not change the SUM of the solutions. I just want to show all of the possible considerations)

If x = 1, then we have: 1^(2(1²) + 4(1) – 6) = 1^(1² + 8(1) + 6)
Simplify: 1^0 = 1^15
Evaluate: 1 = 1
So, x = 1 is another solution to the equation

If x = -1, then we have: (-1)^[2(-1)² + 4(-1) – 6] = (-1)^[(-1)² + 8(-1) + 6]
Simplify: (-1)^(-8) = (-1)^(-1)
Evaluate: 1 = -1
So, x = -1 is NOT a solution to the equation

Now let's assume that x ≠ 0, x ≠ 1 and x ≠ -1 and look for other x-values that satisfy the given equation.
Given: x^(2x² + 4x – 6) = x^(x² + 8x + 6)
Since the bases are the same, we can write: 2x² + 4x – 6 = x² + 8x + 6
Rearrange to get: x² - 4x – 12 = 0
Factor to get: (x - 6)(x + 2) = 0
So, x = 6 and x = -2 are also solutions to the equation.

So, the solutions are x = 0, x = 1, x = 6, and x = -2
0 + 1 + 6 + (-2) = 5

_________________
Test confidently with gmatprepnow.com
##### General Discussion
VP
Joined: 07 Dec 2014
Posts: 1229
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

02 Jan 2017, 16:49
1
GMATPrepNow wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

subtract exponent 2 from exponent 1
x^2-4x-12=0
x=6,-2
6+(-2)=4
D
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4235
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

02 Jan 2017, 16:58
Top Contributor
1
gracie wrote:
GMATPrepNow wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

subtract exponent 2 from exponent 1
x^2-4x-12=0
x=6,-2
6+(-2)=4
D

That's close, but you haven't found all of the possible solutions.

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
VP
Joined: 07 Dec 2014
Posts: 1229
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

02 Jan 2017, 18:41
2
1
GMATPrepNow wrote:
gracie wrote:
GMATPrepNow wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

subtract exponent 2 from exponent 1
x^2-4x-12=0
x=6,-2
6+(-2)=4
D

That's close, but you haven't found all of the possible solutions.

Cheers,
Brent

I see it now, should have noted that 1^0=1^15.
Thanks. Good problem.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4235
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

02 Jan 2017, 20:19
Top Contributor
gracie wrote:
I see it now, should have noted that 1^0=1^15.
Thanks. Good problem.

Nice work.
I should also note that x = 0 is another solution (but it doesn't change the answer)

We must also check to see whether x = -1 is a possible solution.
When we plug x = -1 into the equation, we find that it is NOT a solution.

So, the solutions are x = 6, x = -2 and x = 1

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Director
Joined: 31 Jul 2017
Posts: 503
Location: Malaysia
GMAT 1: 700 Q50 V33
GPA: 3.95
WE: Consulting (Energy and Utilities)
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

20 Mar 2018, 10:49
1
1
GMATPrepNow wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

The equation can be written as -

$$x ^{(x-6)(x+2)} = 1$$

L.H.S = R.H.S, when x = 1,6,-2
Hence, Sum = 5
Manager
Joined: 04 Feb 2016
Posts: 59
Location: India
Concentration: Technology, Marketing
GMAT 1: 650 Q48 V32
GPA: 4
WE: Sales (Computer Software)
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

24 Mar 2018, 02:24
Best approach would be to take log on both sides.

logx[(x-6)(x+2)] =0
_________________
GMAT1 650 Q48 V32.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4235
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

08 May 2018, 13:11
Top Contributor
srinjoy1990 wrote:
Best approach would be to take log on both sides.

logx[(x-6)(x+2)] =0

Unfortunately, you won't have a calculator (or a log table for that day) on test day.
That said, if you did have a calculator, what values would you plug in for x?

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Manager
Joined: 04 Feb 2016
Posts: 59
Location: India
Concentration: Technology, Marketing
GMAT 1: 650 Q48 V32
GPA: 4
WE: Sales (Computer Software)
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

08 May 2018, 13:24
GMATPrepNow wrote:
srinjoy1990 wrote:
Best approach would be to take log on both sides.

logx[(x-6)(x+2)] =0

Unfortunately, you won't have a calculator (or a log table for that day) on test day.
That said, if you did have a calculator, what values would you plug in for x?

Cheers,
Brent

we need not take values thats the point, we can solve the equation by taking log.

you have something like this,

logx(x^2-4x-12) = 0.

logx =0 | x^2-4x-12=0

so, roots are, x=1 , and the solution to the equation x^2-4x-12=0.

so, 5 (1+4).
_________________
GMAT1 650 Q48 V32.
Senior Manager
Joined: 31 May 2018
Posts: 461
Location: United States
Concentration: Finance, Marketing
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

26 Jun 2018, 05:09
2
GMATPrepNow wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

A) -4
B) -3
C) 3
D) 4
E) 5

* Kudos for all correct solutions

x^(2x² + 4x – 6) = x^(x² + 8x +6)
2x² + 4x – 6 = x² + 8x +6
x² - 4x - 12 = (x+2)(x-6)
from here x = -2 , 6
here x = 1 will also satisfy the equation
so -2+6+1 = 5
Intern
Joined: 18 Aug 2017
Posts: 2
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x  [#permalink]

### Show Tags

06 Nov 2019, 10:59
x = 1,6,-2
Sum = 5.
Hence E.

Posted from my mobile device
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x   [#permalink] 06 Nov 2019, 10:59
Display posts from previous: Sort by