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

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What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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02 Jan 2017, 13:45
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Difficulty:

95% (hard)

Question Stats:

18% (01:52) correct 82% (01:32) wrong based on 223 sessions

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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

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Brent Hanneson – Founder of gmatprepnow.com

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Joined: 12 Sep 2015
Posts: 2633
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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03 Jan 2017, 11:20
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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

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Brent Hanneson – Founder of gmatprepnow.com

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Joined: 07 Dec 2014
Posts: 1036
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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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
CEO
Joined: 12 Sep 2015
Posts: 2633
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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02 Jan 2017, 16:58
Top Contributor
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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
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Brent Hanneson – Founder of gmatprepnow.com

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Posts: 1036
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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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.
CEO
Joined: 12 Sep 2015
Posts: 2633
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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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
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Brent Hanneson – Founder of gmatprepnow.com

Senior Manager
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Posts: 374
Location: Malaysia
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What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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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

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Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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

logx[(x-6)(x+2)] =0
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GMAT1 650 Q48 V32.

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Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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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
_________________

Brent Hanneson – Founder of gmatprepnow.com

Manager
Joined: 04 Feb 2016
Posts: 71
Location: India
Concentration: Technology, Marketing
GPA: 3.7
WE: Sales (Computer Software)
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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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).
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Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x [#permalink]

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26 Jun 2018, 05:09
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

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
Re: What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x   [#permalink] 26 Jun 2018, 05:09
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