What is the sum of all possible solutions of the equation |x + 4|^2 -

|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

Correct me:

I solved from where the author of the problem left it. that is:
y = -2 or 12
Hence, considereding + values of |x+4|, i.e. x+4 = -2 or 12, which gives us x = -6 or 8

Considering - values of |x+4|, i.e. -x-4 = -6 or 4, which gives us x = -2 or 8.

Sum of all, -6+8-2+8 = 8.

IMO -16.

Take y = |x+4 | and solve for y, then solve for |x+4| , we get x=-16,8,-2,-6, sum = -16.
OA?

Economist the problem is that -2 and -6 doesn't satisfy the equation. Thus only two values of x are left -16 and 8: -16+8=-8.

Consider this:
|x + 4|^2 - 10|x + 4| = 24
Solve for \(|x+4 |\) --> \(|x+4 |=12\) OR \(|x+4 |=-2\), BUT as absolute value never negative thus -2 is out. Solving \(|x+4 |=12\) --> \(x_1=8\) or \(x_2=-16\) --> \(x_1+x_2=8-16=-8\).

Hope it's clear.

|x+4| = y
gives y^2 -10y -24 = 0

y = -2 and 12

|x+4| = 12 gives x = 8 and -16.

sum is -8.

Hi Buneuel,

Please help me with the basic understanding of the mod probs

when we have

|x+4|= |x-5|

We can two solutions

x+4= x-5
and
x+4=-x+5


but in this problem why do we check for ranges. I mean what s the step by step approach to attack a modulus question?
a few examples would be grateful

Let's try this with number line.
|x+4| = y ==> y^2-10y-24=0 ==> y = 12 or y = -2
Substitute the value of y
we have
|x+4|=12 or |x+4|= -2
Hmm.. can mod be a negative number? NO ==> Eliminate |x+4|= -2

Now we are left only with |x+4|=12
Lets draw a number line
.................................|x+4|.................................
<------------------------------------------------------------------------>
-16..............................(-4)................................8

Thus, two possible roots are -16 and +8
Sum of roots => -16+8=-8

Answer Is D .... But I think this is a 700 level question not 600- 700 level


so , lets start,

| x+4 | ^2 - 10 |x+4 | =24 , Lets put letter t rather than the term | x+4 | in the equation and solve the equation :

so, we have : t ^2 -10 t = 24 or : t^2 -10 t -24=0 OR : (t-12 ) ( t+2) = 0 and from here we get two values for t : t = 12 & -2


So , we have TWO cases : the First case : | x+4 | = 12 and the second case : | x+ 4 | = -2 BUT here notice that the second case is REJECTED as LHS is ALWAYS POSSIBLE ( because of absolute value) , so its value CAN NOT be negative . so we have ONLY ONE scenario .


HERE : | x+4 | =12 OR : x+4 = +/- 12 , so we have TWO scenarios ; Scenario 1) : x+4 = +12 so X= 8 and Scenario 2) : x+4 =-12 SO X= -16

NOW the problem HAS NOT STILL FINISHED !! because we have to check whether two values can be confirm in the equation or NOT..

LETS consider the solution -16 : |-16 + 4 | ^2 -10 |-16+4 | =24 or : 12 ^2 -10*12 = 144-120 =24 so confirms with the RHS ( 24 ) , so accepted

NOW 8 : |8+4| ^2 -10 |8+4| = 12 ^2 -10 *12 = 144 -120 = 24 Confirms with RHS , so accepted,

NOW the sum of all possible solutions = -16 +8 = -8 ANSWER D...

Hi All,

When it comes to "layered" questions, you have to be careful about the approach that you take. The more complex an approach is, the more likely you are to make a mistake, miss a detail or do a calculation that is incorrect.

Here, we have:

|X+4|^2 - 10|X+4| = 24

This is certainly a complex looking calculation, but it IS based on some simple ideas and rules. Rather than take a calculation-heavy approach, let's break this into 'pieces' and talk through what each piece means...

First, I'm going to rewrite the equation:

|X+4|^2 = 24 + 10|X+4|

This tell us that....

|X+4|^2 is exactly 24 "bigger" than 10|X+4|

Next, let's compare pieces...

|X+4|^2 = (|X+4|)(|X+4|)

(|X+4|)(|X+4|) is 24 "bigger" than (10)(|X+4|)

Compare the two products....they each have a (|X+4|) a term. The difference of 24 must be based on the OTHER terms...

|X+4| MUST be > 10

....but probably not that much bigger, since the difference in the overall calculation is just 24.

So.....what happens in this equation: |X+4|^2 = 24 + 10|X+4|

When.....X = 7......
121 = 24 + 110???? This is not correct (121 does NOT = 134)

When....X = 8.....
144 = 24 + 120? This IS correct (144 = 144)

When....X = 9....
169 = 24 + 130??? This is not correct (169 does NOT = 154)

As X gets bigger, we can see that the calculation will NOT be equal. This means that |X+4| MUST = 12 and that ONE of the solutions is X=8. Since we're dealing with an absolute value, we have 2 equations to solve:

X+4 = 12
X = 8

X+4 = -12
X = -16

So the 2 solutions are X=8 and X = -16. There are NO other options.

The prompt asks for the sum of the solutions: -16 + 8 = -8

Final Answer:
GMAT assassins aren't born, they're made,
Rich

Hi all,

Please comment on this:

|x+4|^2 - 10|x+4|=24
i.e. |x+4|(|x+4|-10)=24
i.e. |x+4|=24 or |x+4|-10=24
i.e. |x+4|=24 or |x+4|=34
i.e. x+4=24 or x+4=-24 or x+4=34 or x+4=-34
i.e. x=20 or x=-28 or x=30 or x=-38

So all possible values = 20-28+30-38 = -16 = ANSWER: A

TO

Hi thorinoakenshield,

Your solutions are actually really easy to check. When you plug each of those 4 values in for X, does the equation "balance out?"

eg. IF.....X = 20

Does |20+4|^2 - 10|20+4| = 24?

If it does NOT balance out (re if the calculation does NOT equal 24), then X=20 is NOT a solution.

Now, check the others.

GMAT assassins aren't born, they're made,
Rich

MAGOOSH OFFICIAL SOLUTION:

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

Given that |x + 4|^2 - 10|x + 4| = 24 and we need to find the sum of all possible solutions of the equation

Assume |x+4| = A, we will first solve for A and then use the value of A to solve for |x+4|

=> \(A^2\) - 10A = 24
=> \(A^2\) - 10A - 24 = 0
=> \(A^2\) - 12A + 2A - 24 = 0
=> A*(A-12) + 2*(A-12) = 0
=> (A-12)*(A+2) = 0
=> A = -2, 12

Now, A = |x+4| => A ≥ 0 (as Absolute value of any number can never be negative)

=> Only possible value of A is 12

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

=> Sum of all possible solutions of the equation = 8 + (-16) = -8

So, Answer will be D
Hope it helps!

Watch the following video to learn How to Solve Absolute Value Problems

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