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

##### This topic has expert replies
Legendary Member
Posts: 1486
Joined: 01 Mar 2018
Followed by:2 members

### 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

00:00

A

B

C

D

E

## Global Stats

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

### GMAT/MBA Expert

GMAT Instructor
Posts: 15886
Joined: 08 Dec 2008
Location: Vancouver, BC
Thanked: 5254 times
Followed by:1267 members
GMAT Score:770

### 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