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

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

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29 Oct 2009, 02:30
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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

Let $$x+4=y$$
Now we get two cases,
Case1:
$$y^2-10y-24=0$$
Solving we get -2,12

Case2:
$$-y^2+10y-24=0$$
where we get 6,4
[Reveal] Spoiler: OA

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

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29 Oct 2009, 03:11
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tejal777 wrote:
What is the sum of all roots of the equation
$$|x + 4|^2 - 10|x + 4| = 24?$$

Let $$x+4=y$$
Now we get two cases,
Case1:
$$y^2-10y-24=0$$
Solving we get -2,12

Case2:
$$-y^2+10y-24=0$$
where we get 6,4

This is a good question.

Let me show you how I've solved, maybe it'll help:

APPROACH #1:
We have |x + 4|^2 - 10|x + 4| = 24

|x + 4| flip sign at x=-4, so we should check two ranges:

1. x<=-4
(x+4)^2 + 10x+40=24 ((x+4)^2 as it's square will be the same in both ranges)

x^2+8x+16+10x+16=0 --> x^2+18x+32=0.
Solving for x: x=-16 or x=-2. x=-2 won't work as x<=-4 (see the defined range), hence we have only one solution for this range x=-16.

2. x>-4
(x+4)^2 - 10x-40=24 --> x^2-2x-48=0.
Solving for x: x=-6 or x=8. x=-6 wont work as x>-4, hence we have only one root for this range x=8.

-16+8=-8.

APPROACH #2:

|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 $$x_2=-16$$ --> $$x_1+x_2=8-16=-8$$.

Answer: the sum of all roots of the equation is -8.
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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29 Oct 2009, 06:37
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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.

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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29 Oct 2009, 13:32
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I solve it by replacing |x+4| as k

so k^2-10k-24=0
(k-12)(k+2) = 0
k = 12 or -2
k can not be -2 because it is an absolute value
so
k = 12 = |x+4|
then
x+4 = 12, x = 8
x+4 = -12, x = -16
sum is -8

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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31 Oct 2009, 05:11
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?

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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31 Oct 2009, 21:59
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Economist wrote:
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.
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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07 Oct 2010, 12:19
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mxgms wrote:
jzd wrote:
I solve it by replacing |x+4| as k

so k^2-10k-24=0
(k-12)(k+2) = 0
k = 12 or -2
k can not be -2 because it is an absolute value
so
k = 12 = |x+4|
then
x+4 = 12, x = 8
x+4 = -12, x = -16
sum is -8

I liked that approach, is this always true?

thanks.

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
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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08 Oct 2010, 01:56
-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.
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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08 Dec 2010, 09:38
shrouded1 wrote:
mxgms wrote:
jzd wrote:
I solve it by replacing |x+4| as k

so k^2-10k-24=0
(k-12)(k+2) = 0
k = 12 or -2
k can not be -2 because it is an absolute value
so
k = 12 = |x+4|
then
x+4 = 12, x = 8
x+4 = -12, x = -16
sum is -8

I liked that approach, is this always true?

thanks.

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

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
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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14 Jun 2011, 02:24
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|x+4| = y
gives y^2 -10y -24 = 0

y = -2 and 12

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

sum is -8.
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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17 Apr 2012, 07:04
Hi Buneuel,

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

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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18 Apr 2012, 04:12
shankar245 wrote:
Hi Buneuel,

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

|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.
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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08 Sep 2012, 01:10
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tejal777 wrote:
What is the sum of all roots of the equation
$$|x + 4|^2 - 10|x + 4| = 24?$$

Let $$x+4=y$$
Now we get two cases,
Case1:
$$y^2-10y-24=0$$
Solving we get -2,12

Case2:
$$-y^2+10y-24=0$$
where we get 6,4

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
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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08 Sep 2012, 03:20
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tejal777 wrote:
What is the sum of all roots of the equation
$$|x + 4|^2 - 10|x + 4| = 24?$$

Let $$x+4=y$$
Now we get two cases,
Case1:
$$y^2-10y-24=0$$
Solving we get -2,12

Case2:
$$-y^2+10y-24=0$$
where we get 6,4

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 $$x$$...
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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25 Sep 2012, 10:42
tejal777 wrote:
What is the sum of all roots of the equation
$$|x + 4|^2 - 10|x + 4| = 24?$$

24 ends with 4 and $$10|x+4|$$ ends with 0. So $$|x+4|^2$$ should end with 4.
Options below 0 are out because of the absolute value.
Let's take the squares ending with 4: $$2^2; 8^2; 12^2; 18^2$$ etc...

We find $$12^2 - 10*12 = 24$$.

From here $$x_1=8$$ and $$x_2=-16$$, and the sum: -8.

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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06 Dec 2012, 03:12
It also took me time to understand how to get solutions for absolute values. But thanks to GMATClub...

Here is a detailed explanation on how you could solve for the roots http://burnoutorbreathe.blogspot.com/2012/12/how-to-get-solution-for-absolute-values.html

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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03 Feb 2013, 19:02
Have a question. There are two scenarios -
1. x<=-4

2. x>-4

How do we determine whether to include equal sign (=) in first equation or second equation or does it not matter and can be included in any?

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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04 Feb 2013, 04:12
Apex231 wrote:
Have a question. There are two scenarios -
1. x<=-4

2. x>-4

How do we determine whether to include equal sign (=) in first equation or second equation or does it not matter and can be included in any?

It does not matter in which range you include 4.
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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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03 Jul 2013, 07:43
tejal777 wrote:
What is the sum of all roots of the equation
$$|x + 4|^2 - 10|x + 4| = 24?$$

Let $$x+4=y$$
Now we get two cases,
Case1:
$$y^2-10y-24=0$$
Solving we get -2,12

Case2:
$$-y^2+10y-24=0$$
where we get 6,4

dude here the key
remember Bodmas childhood rule

now keep lx+3 l as a k and re write equation we get k = 12 or k= -2 and then now substitute the mod value
and remember mod can be negative or positive, as we dont know x and we are finding all possible values
we get 8 and -16 once and also we get -2 and -6 i guess so now add them all

wish u a very good luck and make a wish for me too

logic and basic = magic in gmat

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 - [#permalink]

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03 Jul 2013, 14:51
What is the sum of all roots of the equation: |x+4|^2 - 10|x+4| = 24

I. x+4 = y

|y|^2 - 10|y| = 24
y^2 - 10y = 24
y^2 - 10y -24 =0
(y-12)(y+2)
y=12, y=-2
OR
y^2 - (-10y) = 24
y^2 + 10y -24 =0
(y+12)(y-2)
y=-12, y = 2

Y = 10 or Y=-10
INSUFFICIENT

II. y^2 + 10y - 24 = 0
This tells us nothing about X in the stem.

I+II) This validates one of the two solutions available for |x+4|^2 - 10|x+4| = 24 (when we know what x+4 is from #1)
SUFFICIENT

(C)

(I am a bit confused, is this a DS question?)

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Re: What is the sum of all possible solutions of the equation |x + 4|^2 -   [#permalink] 03 Jul 2013, 14:51

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