## What is the sum of all possible solutions of the equation $$|x + 4|^2 - 10|x + 4| = 24?$$

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### What is the sum of all possible solutions of the equation $$|x + 4|^2 - 10|x + 4| = 24?$$

by Gmat_mission » Sun Sep 12, 2021 8:57 am

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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

Source: Magoosh

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### Re: What is the sum of all possible solutions of the equation $$|x + 4|^2 - 10|x + 4| = 24?$$

by [email protected] » Sun Sep 12, 2021 12:20 pm
Gmat_mission wrote:
Sun Sep 12, 2021 8:57 am
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

Source: Magoosh
|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