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

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.

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.

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

y = -2 and 12

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

sum is -8.

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

|x+4| can expand in two ways: if x<=-4 then |x+4|=-(x+4) and if x>-4 then |x+4|=x+4. So, we expand |x+4| for |x + 4|^2 - 10|x + 4| = 24 according to this and then solve for x.

Solution in this post might be easier to understand: what-is-the-sum-of-all-roots-of-the-equation-85988.html#p645659

For basic understanding check Absolute Value chapter of Math Book: math-absolute-value-modulus-86462.html

DS question on absolute values: search.php?search_id=tag&tag_id=37
PS question on absolute values: search.php?search_id=tag&tag_id=58

Hope it helps.

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

Not sure what part you are questioning about
(1) You can always do a variable switch in an equation (|x+4|=y)
(2) |Any expression| is always greater than or equal to 0

Case 1: You mean \(y\geq{0}\), right? Because \(|x+4|=y\) only if \(y\) is non-negative.
Only \(y=12\) is acceptable. From \(|x+4|=12\) we obtain \(x=8\) and \(x=-16.\)

Case 2: Now \(y<0,\) so \(|x+4|=-y\). But \(|x+4|^2=(-y)^2\) is still \(y^2\), doesn't matter that \(y\) is negative!
Your equation should be \(y^2+10y-24=0,\) solutions \(2, -12\). Now only \(-12\) is acceptable (\(y\) must be negative), and we obtain the same solutions as in Case 1.

It would have been better to denote \(|x+4|=y\geq{0}\) (see other posts above). Then \(|x+4|^2=y^2\), and for the quadratic equation \(y^2+10y-24=0\) you choose only the non-negative root, then find \(x\)...

Here's how I did it-

let |x+4| = y

|X+4| ^2 - 10|x+4|= 24 can be written as y^2-10y-24=0

solving for y we get y = -2 or 10

since modulus value cannot be negative, the only possible solution is |x+4| = 10


|x+4| = 10
when x+4>= 0 x+4=10 => x= 6
when x+4<0 -x-4=10 => x= -14


-14 + 6 = 8

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:

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?

-8 for me. Once you solve the QE you get |x+4| = 6 or -4. -4 is not possible so take the case |x+4| = 6 which means x = -10 or 2. So the sum is -8.

Shrouded: can we do this question by the approach you have mentioned in the walker post. .i.e |x-a|<b =>
a-b<x<a+b or this approach is for specific questions.
Thanks

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