What is the product of all possible solutions of the equation |x + 2|^

Question

What is the product of all possible solutions of the equation  |x + 2|^2 - 5|x + 2| = -6 ?

A. -20
B. -5
C. -4
D. 0
E. 20

Bunuel · Accepted Answer

OFFICIAL EXPLANATION: 
 What is the product of all possible solutions of the equation  |x + 2|^2 - 5|x + 2| = -6 ? 
 
A. -20
B. -5
C. -4
D. 0
E. 20

Denote  |x+2|  as  a

We get  a^2 - 5a = -6

Solving quadratics gives  a =3  or  a=2

Thus,  |x+2|=3  or  |x+2|=2

Further solving gives,  x=1 ,  x=-5 ,  x=0 , or  x=-4 .

The product of those values is 0.

Answer: D

exc4libur · Answer

|x + 2|^2 - 5|x + 2| = -6…x^2+4+4x-5|x+2|=-6…x^2+10+4x-5|x+2|=0 if:x+2>0…x>-2…then:|x+2|=positive x^2+10+4x-5|x+2|=0…x^2+10+4x-5x-10=0…x^2-x=0…x(x-1)=0 since:x>-2…then:x=(0,1)=valid.solutions if:x+2<0…x<-2…then:|x+2|=negative x^2+10+4x-5|x+2|=0…x^2+10+4x-5(-x-2)=0…x^2+9x+20=0…(x+4)(x+5)=0 since:x<-2…then:…x=(-4,-5)=valid.solutions product(0,1,-4,-5)=0 Ans. (D)

GMATinsight · Answer

Let,  |x + 2| = a

i.e. equation  |x + 2|^2 - 5|x + 2| = -6  becomes  a^2 - 5a = -6

i.e.  a^2 - 5a +6 = 0

i.e.  a = 3, 2

If  a = 3 = |x + 2|  then  x = 1 or -5

If  a = 2 = |x + 2|  then  x = 0 or -4

Product of all possible values of  x = (1)*(-5)*(0)*(-4) = 0

Answer: Option D

Euphor1a · Answer

Hey Insight,

Hope you're well.

In a question like this, why can't I multiply out the brackets to make a quadratic equation then solve for x? For example, once I multiplied and simplified the above, I deduced: x^2 + 4x - 5 = 0, where x = -5 or 1. Therefore, the answer would be -5.

May you point out the flaw in my reasoning please?

Thanks in advance.

GMATinsight · Answer

It's not bracket (parenthesis) it's modulus sign which considers absolute value i.e. the part inside may be positive as well as negative. You have skipped the possibility of the x+2 to be negative when you treat it like a bracket.

KarishmaB · Answer

We always say that you need to keep an eye on the options from the beginning. Here, I see an equation with absolute values and we are looking for its solutions. 
The product of the solutions look like easy numbers (I think -4, 5 etc).
I notice 0 and the first thing I do here is check for 0. Does x = 0 work? If yes, the product will be 0 no matter what the other values of x are.
x = 0 works! 
We are done.

Answer (D)

Archit3110 · Answer

What is the product of all possible solutions of the equation |x+2|2−5|x+2|=−6|x+2|2−5|x+2|=−6?

A. -20
B. -5
C. -4
D. 0
E. 20

Let |x+2|= Y
y^2-5y+6=0

solve for y = 2,3

since |x+2|= Y
we get values, 0,1,2,3 
product would be 0 hence D

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

CEdward · Answer

So it looks like in this instance if you plug in 0, it works. But doing that presupposes that x = 0. Isn't the question asking what is the product of xy?

Actually...so x = 0 ... xy = 0.

Champer21 · Answer

Can somebody please show me how to insert the values of 3 and 2, to get the the four values of x?

I mean this step:

If a=3=|x+2|a=3=|x+2| then x=1or−5x=1or−5

If a=2=|x+2|a=2=|x+2| then x=0or−4

Bunuel · Answer

We denoted |x + 2| as a, and then got that a = 3 or a = 2. Hence, |x + 2| = 3 or |x + 2| = 2.

If |x + 2| = 3, then either x + 2 = 3 or x + 2 = -3. Thus, x = 1 or x = -5.
If |x + 2| = 2, then either x + 2 = 2 or x + 2 = -2. Thus, x = 0 or x = -4.

Hope it's clear.

Champer21 · Answer

Thank you very much Bunuel. It is not normal how fast you reply.
Only the best for you!

unraveled · Answer

What is the product of all possible solutions of the equation  |x + 2|^2 - 5|x + 2| = -6 ?

A. -20
B. -5
C. -4
D. 0
E. 20

Since its a polynomial of degree two, we will get two answers - unless we have zero. However, it is absolute value that gets two values so any variable inside the absolute would have four values - unless we have zero.

Let's say y = |x+2|
Therefore,
 y^2 - 5y = -6 
gives y = 3 and 2

|x+2| = 3 gives x = 1 and -5
|x+2| = 2 gives x = 0

Now, this gives product as zero.

Answer D.

Anu2021 · Answer

Is it wrong to treat the absolute value like parentheses?
I considered it as parentheses and got - 
x^2 + 4 + 4x -5x - 10 = -6
x^2 -x = 0 
x(x-1) = 0

Therefore, x = 1 and x = 0. 
On multiplication of the roots 1 and 0 , you get 0. 
ans (D).

Is this approach wrong? If yes, why?

Bunuel · Answer

Absolute value and parentheses are not the same.

Absolute Value 
 
    Theory   
 Math: Absolute value (Modulus) 
 Absolute Value: Tips and hints 
 Properties of Absolute Values on the GMAT 
 Absolute modulus : A better understanding 
 GMAT Question Patterns: Abolute Value 
 The Reason Behind Absolute Value Questions on the GMAT 
 
    Questions   
 Inequality and absolute value questions from my collection 
 The E-GMAT Question Series on ABSOLUTE VALUE 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

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What is the product of all possible solutions of the equation |x + 2|^

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