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

HI Bunnel,

For equation x^2-10x-24

We can get two roots (12,-2) it works fine we get value as -8

but for above quadratic equation we can also get 6 and -4 as root and it will give us different result.

Please clarify.

Thanks

x=6 and x=-4 are NOT the roots of x^2-10x-24=0.

While the answer remains the same, shouldn't 1. be x<4 and 2. X>=-4 ?

You can include = sign in either of the ranges.

What is the sum of all roots of the equation
\(|x + 4|^2 - 10|x + 4| = 24?\)


\(|x + 4|^2 - 10|x + 4| = 24\)
Or, \(|x + 4|^2 - 10|x + 4| - 24 = 0\)
Or,\(|x + 4|^2 - 10|x + 4| +5^2 -7^2=0\)
Or, \((|x + 4|+5)^2 -7^2=0\)
Or, \((|x + 4| + 5 +7)(|x + 4| + 5 -7)=0\)
Or, \((|x + 4| +12)(|x + 4|- 2) =0\)
Discarding the negative root -12,
\(|x + 4| = 2\)
or, roots of the equation are\(-6\) and\(-2\) => sum of roots \(=-8\)

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...

I ended up with ANS A (-16)...

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

First case: the positive values:
(x + 4)^2 - 10(x + 4) = 24
x^2 + 4^2 + 2(4x) - 10x - 40 = 24
x^2 + 8x - 10x = 24 + 40 - 16
x^2 - 2x = 48
x^2 - 2x - 48 = 0
(x+6) * (x-8) = 0
x=-6 or x=8

Substituting both into the equation we get 24. So, both solutions should be accepted.

Second case: the negative values:
(-x - 4)^2 - 10(-x - 4) = 24
(-x)^2 + (-4)^2 + 2(4x) +10x + 40 =24
x^2 + 8x + 10x = 24 - 40 - 16
x^2 + 18x = -32
x^2 + 18x + 32 = 0
(x+2) * (x+16) = 0
x=-2 or x=-16

Substituting both into the equation we get 24. So, both solutions should be accepted.

Then, -6-16-2+8 = -16

Unless I wasn't supposed to use (a+b)^2 = a^2 + b^2 + 2ab

So....???

hi pacifist,
you are correct in your approach .
just a point '(-x - 4)^2 - 10(-x - 4) = 24' the coloured portion can be kept positive as it is to the power of 2...
lesser complicated the signs , lesser the chance of creating mistake with signs..

+1 for A. The four possible solutions are 8, -6, -16, -2

Thank you chetan2u. I know what you mean with the signs. Unfortunately I am extremely bad with calculations. I need to do every single step writen before I decide to simplify it because I always miss sth. Even if it is sth as simple as 5+x=7, I need to do this x = 7-5, instead of directly thinkging about it. Otherwise I can very easily end up with -2. It is a sickness really!

But, I have done this mistake over and over and missed difficult questions for such reasons, so I am not taking the chances anymore!

[quote="pacifist85"]I ended up with ANS A (-16)...

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

First case: the positive values:
(x + 4)^2 - 10(x + 4) = 24
x^2 + 4^2 + 2(4x) + 10x +40 = 24
x^2 + 8x - 10x = 24 + 40 - 16
x^2 - 2x = 48
x^2 - 2x - 48 = 0
(x+6) * (x-8) = 0
x=-6 or x=8

Substituting both into the equation we get 24. So, both solutions should be accepte

Second case: the negative values:
(-x - 4)^2 - 10(-x - 4) = 24
(-x)^2 + (-4)^2 + 2(4x) +10x + 40 =24
x^2 + 8x + 10x = 24 - 40 - 16
x^2 + 18x = -32
x^2 + 18x + 32 = 0
(x+2) * (x+16) = 0
x=-2 or x=-16

Substituting both into the equation we get 24. So, both solutions should be accepted.

Then, -6-16-2+8 = -16

Unless I wasn't supposed to use (a+b)^2 = a^2 + b^2 + 2ab

So....???[/quot


Dear Pacifist85,

I think once again the answer is D and unfortunately you made mistake in your calculation because if even we consider your approach true , you have made a mistake , here is the explanation:

IN the first case , you get two solutions (8 , -6) , BUT in order to confirm roots, you must plug -in roots in THE ORIGINAL EQUATION NOT IN THE CASE WHICH YOU MADE,

SO, if you plug-in the 2 roots in the original ( by using the absolute value properties ) you will see that ONLY the ROOT 8 can be confirm and -6 will not confirm . so here ONLY 8 is valid,

SIMILARLY, in the second case, only -16 will be confirm and -2 will not confirm

SO, by adding the TWO CONFIRMED ROOTS (-16, 8 ) together , we get -16+8=-8 , so 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

This was a good question for me to realise some things about absolute values.
I now think that indeed the answer should be D and not A.

However, I wanted to ask sth abouth this methodology I used above for for absolute values (so reversing the signs in the absolute values instead of in the result of the equation).

Is it correct or should we actually not be reversing signs inside absolute values and instead reversing the sign of the result? What is the difference in one or the other?

Because, in what I did above, I didn't use the initial equation (|x + 4|^2 - 10|x + 4| = 24) to test the roots I found, but the equation I used to produce the roots. Which I think is correct, because it is indeed different than the initial one.

However, testing the roots I found from each follow up equation I created, in the same equations, showed that these roots should be accepted. On the other hand, testing them in the initial equation, while keeping their respective signs, prooved that 2 of the solutions should have been rejected.

I think this has to do with the fact that I reversed the signs of the values inside the absolute values. Also, after finding the roots of the first equation I created (First case: the positive values:
(x + 4)^2 - 10(x + 4) = 24), logically, I should have rejected the negative root (-6), as I was already assuming x to be positive. In the second one though, both were negative, so it is not as easy to see that one of them should be rejected.

Sorry for the long text!

MAGOOSH OFFICIAL SOLUTION:

Why is it not (x-4) (x-6)=0?
this is x^2 - 10x - 24 = 0

and then the answers are 4 and 6.
So, |x+4| = 6, which means x=6-4, = 2
|x+4| = -6, which means x=-6-4, =-10
|x+4| = 4, which means x=0
|x+4| = -4, which means x=-8

Total = -16??