 It is currently 26 May 2019, 12:59 ### 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. # What is the sum of all solutions to the equation  Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS: GRE Instructor Joined: 10 Apr 2015
Posts: 1800
Followers: 58

Kudos [?]: 1686  , given: 8

What is the sum of all solutions to the equation [#permalink]
1
This post received
KUDOS
Expert's post 00:00

Question Stats: 45% (01:10) correct 54% (02:01) wrong based on 11 sessions
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

* Kudos for all correct solutions

Answer:
[Reveal] Spoiler:
5

_________________

Brent Hanneson – Creator of greenlighttestprep.com Sign up for my free GRE Question of the Day emails GRE Instructor Joined: 10 Apr 2015
Posts: 1800
Followers: 58

Kudos [?]: 1686  , given: 8

Re: What is the sum of all solutions to the equation [#permalink]
1
This post received
KUDOS
Expert's post
GreenlightTestPrep wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

* Kudos for all correct solutions

Answer:
[Reveal] Spoiler:
5

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

Answer: 5
_________________

Brent Hanneson – Creator of greenlighttestprep.com Sign up for my free GRE Question of the Day emails

Intern Joined: 12 Aug 2018
Posts: 9
Followers: 0

Kudos [?]: 3 , given: 3

Re: What is the sum of all solutions to the equation [#permalink]
isn't it contradictory since you mentioned that the base cannot be 0,1,or-1 yet you tested X for all these values. Appreciate your clarification. Thanks GRE Instructor Joined: 10 Apr 2015
Posts: 1800
Followers: 58

Kudos [?]: 1686  , given: 8

Re: What is the sum of all solutions to the equation [#permalink]
1
This post received
KUDOS
Expert's post
Runnyboy44 wrote:
isn't it contradictory since you mentioned that the base cannot be 0,1,or-1 yet you tested X for all these values. Appreciate your clarification. Thanks

Be careful; I didn't say that the base cannot be 0, 1 or -1.
I said that, if the base equals 0, 1 or -1, then the rule does not necessarily apply.

For example, let's say we're told that b^x = b^y
Can we conclude that x = y?
No. We can only conclude that x = y IF we are certain that b does not equal 0, 1 or -1.

So, before we can make any conclusions about the exponents being equal, we must first ensure that the base does not equal 0, 1 or -1

Does that help?

Cheers,
Brent
_________________

Brent Hanneson – Creator of greenlighttestprep.com Sign up for my free GRE Question of the Day emails

Intern Joined: 13 Jul 2018
Posts: 33
GRE 1: Q159 V151 GPA: 4
WE: Programming (Computer Software)
Followers: 0

Kudos [?]: 19 , given: 9

Re: What is the sum of all solutions to the equation [#permalink]
Equate the x below to get x=1 and equate the equations above to get the solutions -6, 2.
Intern Joined: 20 Sep 2018
Posts: 15
Followers: 0

Kudos [?]: 1 , given: 0

Re: What is the sum of all solutions to the equation [#permalink]
I did not understand how x= 1 and 0.. Though I got the other two values 6 and -2. Plz explain
Supreme Moderator
Joined: 01 Nov 2017
Posts: 370
Followers: 5

Kudos [?]: 111 , given: 4

Re: What is the sum of all solutions to the equation [#permalink]
Expert's post
Reetika1990 wrote:
I did not understand how x= 1 and 0.. Though I got the other two values 6 and -2. Plz explain

whatever be the exponent, if the base is 0 or 1, answer will always be 0 or 1 respectively except when power is negative..
for example in this equation..

$$x^{2x² + 4x – 6} = x^{x² + 8x +6}$$
x=0...
$$0^{-6}=0^6....undefined=0$$, so 0 may not be a value
x=1
$$1^{2*1^2+4*1-6}=1^{1^2+8*1+6}.......1^0=1^{15}....1=1$$...yes
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Supreme Moderator
Joined: 01 Nov 2017
Posts: 370
Followers: 5

Kudos [?]: 111 , given: 4

Re: What is the sum of all solutions to the equation [#permalink]
Expert's post
GreenlightTestPrep wrote:
GreenlightTestPrep wrote:
What is the sum of all solutions to the equation x^(2x² + 4x – 6) = x^(x² + 8x +6) ?

* Kudos for all correct solutions

Answer:
[Reveal] Spoiler:
5

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)

Hi @GreenlightTestPrep,

excellent question..
I do agree 0 will not make a difference to the solution but 0 may not be a value of x here because one side becomes 0 to the power of -6, a negative number..
$$0^{-6}=\frac{1}{0^6}=\frac{1}{0}$$, an undefined value
_________________

Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html Re: What is the sum of all solutions to the equation   [#permalink] 19 Oct 2018, 23:21
Display posts from previous: Sort by

# What is the sum of all solutions to the equation  Question banks Downloads My Bookmarks Reviews Important topics  Powered by phpBB © phpBB Group Kindly note that the GRE® test is a registered trademark of the Educational Testing Service®, and this site has neither been reviewed nor endorsed by ETS®.