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

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Bunuel
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What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   25 Oct 2018, 03:31
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79% (01:55) correct 21% (02:23) wrong based on 1034 sessions
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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

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Bunuel
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   04 Jul 2019, 06:22
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Bunuel wrote:
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


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
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   25 Oct 2018, 04:43
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Bunuel wrote:
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


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
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   25 Oct 2018, 20:47
GMATinsight wrote:
Bunuel wrote:
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


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


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.
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   25 Oct 2018, 22:22
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Euphor1a wrote:
GMATinsight wrote:
Bunuel wrote:
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


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


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.


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.
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   25 Oct 2018, 22:55
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Bunuel wrote:
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


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)
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   27 Oct 2018, 00:00
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
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   24 Dec 2018, 04:40
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Bunuel wrote:
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


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

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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   07 Nov 2019, 13:46
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Bunuel wrote:
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


\(|x + 2|^2 - 5|x + 2| = -6…x^2+4+4x-5|x+2|=-6…x^2+10+4x-5|x+2|=0\)

\(if:x+2>0…x>-2…then:|x+2|=positive\)
\(x^2+10+4x-5|x+2|=0…x^2+10+4x-5x-10=0…x^2-x=0…x(x-1)=0\)
\(since:x>-2…then:x=(0,1)=valid.solutions\)

\(if:x+2<0…x<-2…then:|x+2|=negative\)
\(x^2+10+4x-5|x+2|=0…x^2+10+4x-5(-x-2)=0…x^2+9x+20=0…(x+4)(x+5)=0\)
\(since:x<-2…then:…x=(-4,-5)=valid.solutions\)

\(product(0,1,-4,-5)=0\)

Ans. (D)
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   24 Jan 2021, 22:36
VeritasKarishma wrote:
Bunuel wrote:
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


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)


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.
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   24 Jan 2024, 07:19
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
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   24 Jan 2024, 07:38
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Champer21 wrote:
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


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.
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   24 Jan 2024, 08:06
Bunuel wrote:
Champer21 wrote:
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


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.


Thank you very much Bunuel. It is not normal how fast you reply.
Only the best for you!
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink] New post   24 Jan 2024, 09:37
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.
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Re: What is the product of all possible solutions of the equation |x + 2|^ [#permalink]
24 Jan 2024, 09:37
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