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

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Math Expert
Joined: 02 Sep 2009
Posts: 50583
What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 02:31
00:00

Difficulty:

5% (low)

Question Stats:

88% (01:21) correct 12% (01:58) wrong based on 57 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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CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2698
Location: India
GMAT: INSIGHT
WE: Education (Education)
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 03:43
1
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 $$x = (1)*(-5)*(0)*(-4) = 0$$

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Intern
Joined: 05 Sep 2018
Posts: 7
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 19:47
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 $$x = (1)*(-5)*(0)*(-4) = 0$$

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?

CEO
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 2698
Location: India
GMAT: INSIGHT
WE: Education (Education)
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 21:22
1
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 $$x = (1)*(-5)*(0)*(-4) = 0$$

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?

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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Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8527
Location: Pune, India
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 21:55
3
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.

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Manager
Joined: 18 Aug 2017
Posts: 183
Concentration: Healthcare, Marketing
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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26 Oct 2018, 23:00
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
Re: What is the product of all possible solutions of the equation |x + 2|^ &nbs [#permalink] 26 Oct 2018, 23:00
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