Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 18 Dec 2013, 22:43

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# What is the product of all the solutions of x^2 - 4x + 6=3

Author Message
TAGS:
Moderator
Joined: 01 Sep 2010
Posts: 1999
Followers: 130

Kudos [?]: 1164 [2] , given: 554

What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  13 Feb 2013, 04:22
2
KUDOS
Expert's post
00:00

Difficulty:

65% (medium)

Question Stats:

51% (02:24) correct 48% (01:17) wrong based on 226 sessions
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8
(B) -4
(C) 2
(D) 4
(E) 8

[Reveal] Spoiler:
Edited the answer choice C from -2 to 2 and the OA. It's C not E.
[Reveal] Spoiler: OA

_________________

KUDOS is the good manner to help the entire community.

Last edited by Bunuel on 06 Jul 2013, 01:19, edited 2 times in total.
Edited the answer choices and the OA.
Director
Joined: 24 Aug 2009
Posts: 514
Schools: Harvard, Columbia, Stern, Booth, LSB,
Followers: 8

Kudos [?]: 297 [1] , given: 241

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  13 Feb 2013, 04:28
1
KUDOS
carcass wrote:
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1|?
(A) -8
(B) -4
(C) -2
(D) 4
(E) 8

If |x - 1|>=0 ---->then the modulus will be equal to (x-1) & roots of the resulting equation will be 2,1
If |x - 1|<0 ---->then the modulus will be equal to (-x+1) & roots of the resulting equation will be 4,1

So the product of all the roots (2,1,4,1) is 8.
_________________

If you like my Question/Explanation or the contribution, Kindly appreciate by pressing KUDOS.
Kudos always maximizes GMATCLUB worth
-Game Theory

If you have any question regarding my post, kindly pm me or else I won't be able to reply

Manager
Joined: 02 Jan 2013
Posts: 53
GMAT 1: 750 Q51 V40
WE: Consulting (Consulting)
Followers: 0

Kudos [?]: 25 [0], given: 1

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  13 Feb 2013, 08:14
In order for |x - 1| to be equal to 1 - x, we would have to have x < 1 . Therefore eliminating your second pair of solutions

You could also verify this by substituting x = 4 inthe original equation, and seeing that this solution DOES NOT fit. The only two solutions are 1 and 2.

ANS: no correct option available
Posted from my mobile device
_________________

Please press "kudo" if this helped you!

Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [7] , given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  13 Feb 2013, 09:13
7
KUDOS
Expert's post
carcass wrote:
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8
(B) -4
(C) 2
(D) 4
(E) 8

If x<1, then |x - 1| = -(x-1)=1-x, so in this case we'll have x^2 - 4x + 6 = 3-(1-x) --> x^2-5x+4=0 --> x=1 or x=4 --> discard both solutions since neither is in the range x<1.

If x\geq{1}, then |x - 1| = x-1, so in this case we'll have x^2 - 4x + 6 = 3-(x-1) --> x^2-3x+2=0 --> x=1 or x=2.

Therefore, the product of the roots is 1*2=2.

_________________
Senior Manager
Joined: 13 Jan 2012
Posts: 297
Weight: 170lbs
GMAT 1: 730 Q48 V42
GMAT 2: 740 Q48 V42
WE: Analyst (Other)
Followers: 6

Kudos [?]: 41 [0], given: 35

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  14 Feb 2013, 10:44
Bunuel wrote:
carcass wrote:
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8
(B) -4
(C) -2
(D) 4
(E) 8

If x<1, then |x - 1| = -(x-1)=1-x, so in this case we'll have x^2 - 4x + 6 = 3-(1-x) --> x^2-5x+4=0 --> x=1 or x=4 --> discard both solutions since neither is in the range x<1.

If x\geq{1}, then |x - 1| = x-1, so in this case we'll have x^2 - 4x + 6 = 3-(x-1) --> x^2-3x+2=0 --> x=1 or x=2.

Therefore, the product of the roots is 1*2=2.

No correct answer among the choices.

Wow! Really? How often do we see this? Who the heck wrote this question?
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [0], given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  15 Feb 2013, 02:43
Expert's post
Bunuel wrote:
carcass wrote:
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8
(B) -4
(C) -2
(D) 4
(E) 8

If x<1, then |x - 1| = -(x-1)=1-x, so in this case we'll have x^2 - 4x + 6 = 3-(1-x) --> x^2-5x+4=0 --> x=1 or x=4 --> discard both solutions since neither is in the range x<1.

If x\geq{1}, then |x - 1| = x-1, so in this case we'll have x^2 - 4x + 6 = 3-(x-1) --> x^2-3x+2=0 --> x=1 or x=2.

Therefore, the product of the roots is 1*2=2.

No correct answer among the choices.

Wow! Really? How often do we see this? Who the heck wrote this question?

How often do we see what?

I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.
_________________
Senior Manager
Joined: 13 Jan 2012
Posts: 297
Weight: 170lbs
GMAT 1: 730 Q48 V42
GMAT 2: 740 Q48 V42
WE: Analyst (Other)
Followers: 6

Kudos [?]: 41 [0], given: 35

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  16 Feb 2013, 11:37
Bunuel wrote:

How often do we see what?

I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

How often do we see no correct answer among the answer choices?
Manager
Joined: 21 Jun 2011
Posts: 70
Followers: 1

Kudos [?]: 5 [0], given: 11

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 03:27
Bunuel wrote:
carcass wrote:
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8
(B) -4
(C) -2
(D) 4
(E) 8

If x<1, then |x - 1| = -(x-1)=1-x, so in this case we'll have x^2 - 4x + 6 = 3-(1-x) --> x^2-5x+4=0 --> x=1 or x=4 --> discard both solutions since neither is in the range x<1.

If x\geq{1}, then |x - 1| = x-1, so in this case we'll have x^2 - 4x + 6 = 3-(x-1) --> x^2-3x+2=0 --> x=1 or x=2.

Therefore, the product of the roots is 1*2=2.

No correct answer among the choices.

I have a question, I do understand that why have you taken the value 1 but I don't understand why have you taken x>=1. Why not simply x>1
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [0], given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 03:35
Expert's post
davidfrank wrote:
Bunuel wrote:
carcass wrote:
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?

(A) -8
(B) -4
(C) -2
(D) 4
(E) 8

If x<1, then |x - 1| = -(x-1)=1-x, so in this case we'll have x^2 - 4x + 6 = 3-(1-x) --> x^2-5x+4=0 --> x=1 or x=4 --> discard both solutions since neither is in the range x<1.

If x\geq{1}, then |x - 1| = x-1, so in this case we'll have x^2 - 4x + 6 = 3-(x-1) --> x^2-3x+2=0 --> x=1 or x=2.

Therefore, the product of the roots is 1*2=2.

No correct answer among the choices.

I have a question, I do understand that why have you taken the value 1 but I don't understand why have you taken x>=1. Why not simply x>1

x could be 1, thus when you consider the ranges you should include this value in either of the range, so we could consider x<1 and x>=1 OR x<=1 and x>1 (you cam include = sign in either of the ranges).

Hope it's clear.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [0], given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 03:48
Expert's post
Bunuel wrote:

How often do we see what?

I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

How often do we see no correct answer among the answer choices?

Never, if it's a proper GMAT question.
_________________
Manager
Joined: 10 Apr 2012
Posts: 228
Location: United States
Concentration: Technology, Other
GMAT Date: 09-30-2013
GPA: 2.44
WE: Project Management (Telecommunications)
Followers: 1

Kudos [?]: 32 [0], given: 274

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 05:16
I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.[/quote]

if |x+2|>=0 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<0 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

Last edited by guerrero25 on 18 Feb 2013, 05:47, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [0], given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 05:32
Expert's post
guerrero25 wrote:
I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

if |x+2|>=-2 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<-2 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

I guess you meant the following:

When x\leq{-2}, then |x+2|=-(x-2).
When x>{-2}, then |x+2|=(x-2).

Complete solution:

x^2 + 4x + 7 = |x + 2| + 3 --> x^2 + 4x + 4 = |x + 2| --> (x+2)^2=|x+2| --> (x+2)^4=(x+2)^2 --> (x+2)^2((x+2)^2-1)=0:

x+2=0 --> x=-2;
OR
(x+2)^2-1=0 --> (x+2)^2=1 --> x=-1 or x=-3.

The product of the roots: (-2)*(-1)*(-3)=-6.

Hope it's clear.
_________________
Manager
Joined: 10 Apr 2012
Posts: 228
Location: United States
Concentration: Technology, Other
GMAT Date: 09-30-2013
GPA: 2.44
WE: Project Management (Telecommunications)
Followers: 1

Kudos [?]: 32 [0], given: 274

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 05:48
Bunuel wrote:
guerrero25 wrote:
I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

if |x+2|>=-2 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<-2 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

I guess you meant the following:

When x\leq{-2}, then |x+2|=-(x-2).
When x>{-2}, then |x+2|=(x-2).

Complete solution:

x^2 + 4x + 7 = |x + 2| + 3 --> x^2 + 4x + 4 = |x + 2| --> (x+2)^2=|x+2| --> (x+2)^4=(x+2)^2 --> (x+2)^2((x+2)^2-1)=0:

x+2=0 --> x=-2;
OR
(x+2)^2-1=0 --> (x+2)^2=1 --> x=-1 or x=-3.

The product of the roots: (-2)*(-1)*(-3)=-6.

Hope it's clear.

thanks ! That was a Typo . I edited the post .
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [1] , given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  18 Feb 2013, 05:52
1
KUDOS
Expert's post
guerrero25 wrote:
I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

if |x+2|>=0 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<0 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans[/quote]

Still not correct. Absolute value cannot be negative, so |x+2| is always more than or equal to zero.
_________________
Intern
Joined: 25 Apr 2013
Posts: 1
Followers: 0

Kudos [?]: 0 [0], given: 1

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  25 Apr 2013, 06:23
Hello everyone,

I am so close to understanding this question, but the one thing I do not understand is why the positive of |x+2| is >= and the negative of |x+2| is just <?

Sorry if its a dumb question

Paul
Intern
Joined: 04 May 2013
Posts: 45
Followers: 0

Kudos [?]: 1 [0], given: 7

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  05 Jul 2013, 12:27
Bunuel wrote:
guerrero25 wrote:
I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

if |x+2|>=-2 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<-2 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

I guess you meant the following:

When x\leq{-2}, then |x+2|=-(x-2).
When x>{-2}, then |x+2|=(x-2).

Complete solution:

x^2 + 4x + 7 = |x + 2| + 3 --> x^2 + 4x + 4 = |x + 2| --> (x+2)^2=|x+2| --> (x+2)^4=(x+2)^2 --> (x+2)^2((x+2)^2-1)=0:

x+2=0 --> x=-2;
OR
(x+2)^2-1=0 --> (x+2)^2=1 --> x=-1 or x=-3.

The product of the roots: (-2)*(-1)*(-3)=-6.

Hope it's clear.

Bunuel,
Can you solve this problem using the other method that you used in the previous problem?

I mean:

If
x >= 0, |x + 2| = x + 2.
This would give the equation: x^2 + 4x + 7 = x + 5.
Roots are -2, and -1

what is the other scenario?
What happens if x < 0?

How do we end up with the roots -3, and -1??
Thaanks
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [0], given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  05 Jul 2013, 12:40
Expert's post
jjack0310 wrote:
Bunuel wrote:
guerrero25 wrote:
I've seen similar question which reads:
What is the product of all the solutions of x^2 + 4x + 7 = |x + 2| + 3 ?
A. -6
B. -2
C. 2
D. 6
E. 12

OA:
[Reveal] Spoiler:
A

Try it. I'll provide solution for this question later, if necessary.

if |x+2|>=-2 then |x+2|= (x+2)

eqation becomes (x+2)(x+1)=0

x=-2,-1

if |x+2|<-2 then |x+2|=-(x+2)

equation becomes (x+2) (x+3) =0
x=-2,-3 ( can't be -2 since x<-2)

product of the solution -

-2*-1*-3= -6 Ans

I guess you meant the following:

When x\leq{-2}, then |x+2|=-(x-2).
When x>{-2}, then |x+2|=(x-2).

Complete solution:

x^2 + 4x + 7 = |x + 2| + 3 --> x^2 + 4x + 4 = |x + 2| --> (x+2)^2=|x+2| --> (x+2)^4=(x+2)^2 --> (x+2)^2((x+2)^2-1)=0:

x+2=0 --> x=-2;
OR
(x+2)^2-1=0 --> (x+2)^2=1 --> x=-1 or x=-3.

The product of the roots: (-2)*(-1)*(-3)=-6.

Hope it's clear.

Bunuel,
Can you solve this problem using the other method that you used in the previous problem?

I mean:

If
x >= 0, |x + 2| = x + 2.
This would give the equation: x^2 + 4x + 7 = x + 5.
Roots are -2, and -1

what is the other scenario?
What happens if x < 0?

How do we end up with the roots -3, and -1??
Thaanks

When x\leq{-2}, then |x+2|=-(x-2). So, in this case we'll have x^2 + 4x + 7 =-(x + 2) + 3 --> x=-3 or x=-2. Both solutions are valid.

When x>{-2}, then |x+2|=(x-2). So, in this case we'll have x^2 + 4x + 7 =(x + 2) + 3 --> x=-2 or x=-1. The first solution is not valid since it's out of the range we consider. The second one is OK.

So, there are 3 valid solutions: x=-3, x=-2 and x=-1.

Hope it's clear.
_________________
Intern
Joined: 04 May 2013
Posts: 45
Followers: 0

Kudos [?]: 1 [0], given: 7

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  06 Jul 2013, 08:09
Sorry. It is not clear.

Can you explain what was wrong with the way I was approaching the problem?

I mean other than the part that you marked red, what was I doing wrong? Do I have to solve theproblem using the solution that you mentioned?

If x > -2, how is |x + 2| = (x - 2)?
Is there an identity that I am missing?
If I plug in, X = -1, |x + 2| = 1, but (x - 2) = -3

Why the discrepancy? What identity am I missing?
Math Expert
Joined: 02 Sep 2009
Posts: 15197
Followers: 2556

Kudos [?]: 15806 [0], given: 1571

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  06 Jul 2013, 08:16
Expert's post
jjack0310 wrote:
Sorry. It is not clear.

Can you explain what was wrong with the way I was approaching the problem?

I mean other than the part that you marked red, what was I doing wrong? Do I have to solve theproblem using the solution that you mentioned?

If x > -2, how is |x + 2| = (x - 2)?
Is there an identity that I am missing?
If I plug in, X = -1, |x + 2| = 1, but (x - 2) = -3

Why the discrepancy? What identity am I missing?

There was a typo:
When x\leq{-2}, then |x+2|=-(x+2)
When x>{-2}, then |x+2|=(x+2).

Absolute value properties:

When x\leq{0} then |x|=-x, or more generally when some \ expression\leq{0} then |some \ expression|={-(some \ expression)}. For example: |-5|=5=-(-5);

When x\geq{0} then |x|=x, or more generally when some \ expression\geq{0} then |some \ expression|={some \ expression}. For example: |5|=5.

For our question, when x>-2 (when x+2>0), |x+2|=x+2.

Hope it's clear.
_________________
Intern
Joined: 04 May 2013
Posts: 45
Followers: 0

Kudos [?]: 1 [0], given: 7

Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  06 Jul 2013, 08:52
Bunuel wrote:
jjack0310 wrote:
Sorry. It is not clear.

Can you explain what was wrong with the way I was approaching the problem?

I mean other than the part that you marked red, what was I doing wrong? Do I have to solve theproblem using the solution that you mentioned?

If x > -2, how is |x + 2| = (x - 2)?
Is there an identity that I am missing?
If I plug in, X = -1, |x + 2| = 1, but (x - 2) = -3

Why the discrepancy? What identity am I missing?

There was a typo:
When x\leq{-2}, then |x+2|=-(x+2)
When x>{-2}, then |x+2|=(x+2).

Absolute value properties:

When x\leq{0} then |x|=-x, or more generally when some \ expression\leq{0} then |some \ expression|={-(some \ expression)}. For example: |-5|=5=-(-5);

When x\geq{0} then |x|=x, or more generally when some \ expression\geq{0} then |some \ expression|={some \ expression}. For example: |5|=5.

For our question, when x>-2 (when x+2>0), |x+2|=x+2.

Hope it's clear.

Got it.

Thanks,

Final question, why are there two possibilities for when x = 0? Is that correct? or a typo?
Re: What is the product of all the solutions of x^2 - 4x + 6=3   [#permalink] 06 Jul 2013, 08:52
Similar topics Replies Last post
Similar
Topics:
x^2 + 4x + 4/x + 1/x^2 + 5= 0 . What is the sum of the roots 12 26 Apr 2006, 07:33
What is the value of x? (1) x^4+x^2+1=(1/x^4+x^2+1) (2) 9 15 May 2007, 18:48
product of all possible solutions 8 02 Jun 2009, 03:59
What are the unique values of b and c in the equation 4x^2 + 2 27 Oct 2010, 11:12
6 4^x + 4 ^-x = 2 What is the value of X 12 07 Feb 2012, 08:56
Display posts from previous: Sort by