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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: 60627
What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 03:31
25
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15% (low)

Question Stats:

77% (01:49) correct 23% (02:16) wrong based on 351 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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Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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25 Oct 2018, 22:55
4
4
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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Karishma
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Math Expert
Joined: 02 Sep 2009
Posts: 60627
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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04 Jul 2019, 06:22
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.

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

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25 Oct 2018, 04:43
1
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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Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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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 $$x = (1)*(-5)*(0)*(-4) = 0$$

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?

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

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

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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
Math Expert
Joined: 02 Sep 2009
Posts: 60627
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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24 Dec 2018, 04:40
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

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Posts: 1100
Location: United States
Re: What is the product of all possible solutions of the equation |x + 2|^  [#permalink]

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07 Nov 2019, 13:46
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

$$|x + 2|^2 - 5|x + 2| = -6…x^2+4+4x-5|x+2|=-6…x^2+10+4x-5|x+2|=0$$

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

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

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

Ans. (D)
Re: What is the product of all possible solutions of the equation |x + 2|^   [#permalink] 07 Nov 2019, 13:46
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