What is the sum of all possible solutions of the equation?

[GMATprofessional21]

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

[Brent@GMATPrepNow]

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) = -8

Answer: D

Cheers,
Brent

[Scott@TargetTestPrep]

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

Solution:

Letting u = |x + 4|, we have:

u^2 - 10u = 24

u^2 - 10u - 24 = 0

(u - 12)(u + 2) = 0

u = 12 or u = -2

Since u = |x + 4|, we have |x + 4| = 12 or |x + 4| = -2. We can reject the second equation since an absolute value can’t be negative. So we consider only the first equation: |x + 4| = 12.

If x + 4 is positive:

x + 4 = 12

x = 8

If x + 4 is negative:

-x - 4 = 12

-x = 16

x = -16

The two possible solutions are 8 and -16, so their sum is 8 + -16 = -8

Answer: D