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by Brent@GMATPrepNow » Sun May 25, 2014 9:09 am
[email protected] wrote:What is the sum of all possible solutions of the equation |x + 4|2 - 10|x + 4| = 24?


-16
-14
-12
-8
-6
Is that equation supposed to have a power of 2, as in: |x + 4|^2 - 10|x + 4| = 24 ??

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by [email protected] » Sun May 25, 2014 9:11 am
Brent@GMATPrepNow wrote:
[email protected] wrote:What is the sum of all possible solutions of the equation |x + 4|2 - 10|x + 4| = 24?


-16
-14
-12
-8
-6
Is that equation supposed to have a power of 2, as in: |x + 4|^2 - 10|x + 4| = 24 ??

Cheers,
Brent
Yes it is actually Brent!
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by Brent@GMATPrepNow » Sun May 25, 2014 9:18 am
[email protected] wrote:What is the sum of all possible solutions of the equation |x + 4|² - 10|x + 4| = 24?


A) -16
B) -14
C) -12
D) -8
E) -6
|x + 4|² - 10|x + 4| = 24
Let's simplify matters by using some u-substitution.

Let u = |x + 4| and then replace |x + 4| with u to get: u² - 10u = 24
Subtract 24 from both sides to get: u² - 10u - 24 = 0
Factor to get: (u - 12)(u + 2) = 0
So, u = 12 or u = -2

Now let's replace u with |x + 4|.
This means that |x + 4| = 12 or |x + 4| = -2

If |x + 4| = 12, then x = 8 or -16
If |x + 4| = -2, then there are NO SOLUTIONS, since |x + 4| will always be greater than or equal to zero.

So, there are only 2 solutions: x = 8 and x = -16
We're asked to find the SUM of all possible solutions
x = 8 + (-16) = [spoiler]-8 = D[/spoiler]

Cheers,
Brent
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by [email protected] » Sun May 25, 2014 9:22 am
Brent@GMATPrepNow wrote:
[email protected] wrote:What is the sum of all possible solutions of the equation |x + 4|² - 10|x + 4| = 24?


A) -16
B) -14
C) -12
D) -8
E) -6
|x + 4|² - 10|x + 4| = 24
Let's simplify matters by using some u-substitution.

Let u = |x + 4| and then replace |x + 4| with u to get: u² - 10u = 24
Subtract 24 from both sides to get: u² - 10u - 24 = 0
Factor to get: (u - 12)(u + 2) = 0
So, u = 12 or u = -2

Now let's replace u with |x + 4|.
This means that |x + 4| = 12 or |x + 4| = -2

If |x + 4| = 12, then x = 8 or -16
If |x + 4| = -2, then there are NO SOLUTIONS, since |x + 4| will always be greater than or equal to zero.

So, there are only 2 solutions: x = 8 and x = -16
We're asked to find the SUM of all possible solutions
x = 8 + (-16) = [spoiler]-8 = D[/spoiler]

Cheers,
Brent
Thanks Brent- Is there any other way to solve this question?
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by theCodeToGMAT » Sun May 25, 2014 9:28 am
|x+4|^2 + 10 * |x + 4| = 24

Let |x+4| = y

y^2 + 10y - 24 = 0

==> y = -12 & 2

So,

|x+4| = -12 INVALID CASE


|x+4| = 2 == -x-4 = 2
==> -2 & -6

So, -2 -6 = -8
Last edited by theCodeToGMAT on Sun May 25, 2014 9:40 pm, edited 1 time in total.
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by Brent@GMATPrepNow » Sun May 25, 2014 9:55 am
[email protected] wrote: Thanks Brent- Is there any other way to solve this question?
I can't think of a faster way. That said, perhaps I'm missing something.

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by Brent@GMATPrepNow » Sun May 25, 2014 9:59 am
theCodeToGMAT wrote:|x+4|^2 + 10 * |x + 4| = 24

Let |x+4| = y

y^2 + 10y - 24 = 0

==> y = -12 & 2

So,

|x+4| = -12 == -x-4=-12
==> x = -16 & 8

|x+4| = 2 == -x-4 = 2
==> -2 & -6

So, -16 +8 -2 - 6 = -16
Hi Rahul,

Looks like you transcribed the question incorrectly.

That said, I should point out a common mistake that you made (above in green).
If |x+4| = -12, then this equation can have NO SOLUTIONS, since it's impossible for the absolute value of anything to equal -12.
So, in general, we can say that the equation |something| = negative value has no solutions.

Cheers,
Brent
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by GMATGuruNY » Sun May 25, 2014 4:21 pm
[email protected] wrote:What is the sum of all possible solutions of the equation |x + 4|² - 10|x + 4| = 24?

-16
-14
-12
-8
-6
|x + 4|² = (x+4)² = x² + 8x + 16.
Replacing |x + 4|² with x² + 8x + 16, we get:
x² + 8x + 16 - 10|x+4| = 24
x² + 8x - 8 - 10|x+4| = 0.

|x+4| must represent a NONNEGATIVE value.

Case 1: x≥-4
In this case, |x+4| = x+4.
To illustrate:
If x=-3, then |x+4| = x+4 = -3+4 = 1.

Case 2: x<-4
In this case, |x+4| = -(x+4) = -x-4.
To illustrate:
If x=-5, then |x+4| = -x-4 = -(-5) - 4 = 1.

Case 1: x≥-4
Replacing |x+4| with x+4, we get:
x² + 8x - 8 - 10(x+4) = 0
x² + 8x - 8 - 10x - 40 = 0
x² - 2x - 48 = 0
(x-8)(x+6) = 0
x=8 or x=-6.
Since the tested range is x≥-4, only x=8 is within the required range.
Thus, x=8.

Case 2: x<-4
Replacing |x+4| with -x-4, we get:
x² + 8x - 8 - 10(-x-4) = 0
x² + 8x - 8 + 10x + 40 = 0
x² + 18 + 32 = 0
(x+16)(x+2) = 0
x=-16 or x=-2.
Since the tested range is x<-4, only x=-16 is within the required range.
Thus, x=-16.

Sum of the solutions = 8 + (-16) = -8.

The correct answer is D.
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