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

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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 06 Jul 2013, 09:53
jjack0310 wrote:
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?


No, that's not a typo: |0|=0=-0.
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 09 Jul 2013, 15:25
What is the product of all the solutions of x^2 - 4x + 6 = 3 - |x - 1| ?
x^2 - 4x + 6 = 3 - |x - 1|
x^2 - 4x + 3 = - |x - 1|

x>1
x^2 - 4x + 6 = 3 - |x - 1|
x^2 - 4x + 3 = - (x-1)
x^2 - 4x + 3 = -x+1
x^2 - 3x +2 = 0
(x-1)(x-2)=0
x=1, x=2
2 falls within the range of x>1
x=2

x<1
x^2 - 4x + 6 = 3 - |x - 1|
x^2 - 4x + 6 = 3 - -(x-1)
x^2 - 4x + 6 = 3 - (-x+1)
x^2 - 4x + 3 = + x - 1
x^2 - 5x + 4 = 0
(x-1)(x-4) = 0
x=1, x=4
Neither 1 or 4 fall within the range of x<1
INVALID

The product is (2)

(C)

(Bunuel, could you explain to me how we know where the greater than or equals sign goes in these problems? I see that in your solution you had x>=1, but I don't know why that is as opposed to say, x<=1)

Thanks!
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 19 Jun 2014, 02:57
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.

Answer: C.


Dear Bunuel,

We do not need to consider two situation of |x - 1|.

As, x^2 - 4x + 6 = 3 - |x - 1| <=> x^2 - 4x + 3 = - |x - 1| <0 => 1<x (<3) => x^2 - 4x + 3 = x-1 => x^2 - 5x + 4 = 0 => 2 solutions
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What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 11 Aug 2015, 18:03
paul16 wrote:
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


I have this question too. Typically it doesnt matter if we use the +ve or -ve absolute value function since the value is zero anyways. But for this case, we need to use the value to include/exclude values, so is >= and < the standard?
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 11 Aug 2015, 18:15
ggurface wrote:
paul16 wrote:
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


I have this question too. Typically it doesnt matter if we use the +ve or -ve absolute value function since the value is zero anyways. But for this case, we need to use the value to include/exclude values, so is >= and < the standard?


1 thing, the text in red is not always true. |x| can be any value other than 0 as well.

Coming back to your question, the reason we include "=" with ">" sign because |x| = x for x\(\geq\)0 while |x| =-x for x<0

The convention is to always include "+" with ">" and not with "<" as the NATURE of |x| remains the same for x\(\geq\) 0, while the nature changes for |x| (we need to put a negative infront of 'x') when x<0.

For example, |4| = 4 or |0| = 0 but |-5| = -(-5) = 5

Hope this helps.
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 10 Jun 2016, 17:54
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:

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: \((-2)*(-1)*(-3)=-6\).

Answer: A.

Hope it's clear.



Hi Bunuel,

When I tried to follow the same method I get different answer for the below question.

x^2 + 4x + 7 = |x + 2| + 3.


When x <= 0 then |x + 2| = - (x+2 ) = -x - 2.

Then x^2 + 4x + 7 = - x - 2 + 3.

=> x^2 + 4x + 7 = - x + 1.
=> x^2 + 5x + 6 = 0 => ( x +2 ) ( x + 3) = 0 => x = -2 and - 3 . ( Since x < = 0 both the numbers are possible ).


When x > 0 then |x + 2| = x + 2.

Then x^2 + 4x + 7 = x + 2 + 3.
=> x^2 + 4x + 7 = x + 5.
=> x^2 + 3x + 2 = 0 .
=>( x + 2 ) ( x + 1 ) = 0.
x = - 2 and - 1 ( Since x > 0 both numbers doesn't fit in the range ).

Then suitable numbers would be -2 and -3 and the product should be 6.

Please clarify me if I am missing anything...
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 11 Jun 2016, 00:43
msk0657 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:

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: \((-2)*(-1)*(-3)=-6\).

Answer: A.

Hope it's clear.



Hi Bunuel,

When I tried to follow the same method I get different answer for the below question.

x^2 + 4x + 7 = |x + 2| + 3.


When x <= 0 then |x + 2| = - (x+2 ) = -x - 2.

Then x^2 + 4x + 7 = - x - 2 + 3.

=> x^2 + 4x + 7 = - x + 1.
=> x^2 + 5x + 6 = 0 => ( x +2 ) ( x + 3) = 0 => x = -2 and - 3 . ( Since x < = 0 both the numbers are possible ).


When x > 0 then |x + 2| = x + 2.

Then x^2 + 4x + 7 = x + 2 + 3.
=> x^2 + 4x + 7 = x + 5.
=> x^2 + 3x + 2 = 0 .
=>( x + 2 ) ( x + 1 ) = 0.
x = - 2 and - 1 ( Since x > 0 both numbers doesn't fit in the range ).

Then suitable numbers would be -2 and -3 and the product should be 6.

Please clarify me if I am missing anything...


The transition point (value) must be the point for which the expression in modulus changes its sign. For |x + 2| it's -2 not 0. So, you should consider the ranges when x<=-2 and when x>-2. Check highlighted text in my solution.
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 25 Jul 2016, 20:52
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.

Answer: C.



Hi Bunuel,

I don't know whether my alternative method is right to solve this question:

1) I tried to create a square:
X^2-4X+6=3-|X-1|
X^2-4X+3=-|X-1|
X^2-2X+1-2X+2=-|X-1|
(X-1)^2-2(X-1)+|X-1|=0

2) replace X-1 with Y
Y^2-2Y+|Y|=0

3) if Y>0, the equation will be:
Y^2-Y=0, Y(Y-1)=0, then X-1=0 or X-1=1
So we have X=1 or X=2

4) if Y=0, X-1=0, X=1

4) if Y<0, all in the left side are positive, thus cannot resulting in 0

Consequently, X=1 or X=2, the product is 2.

Thanks,
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 25 Jul 2016, 21:21
mod can be taken + and -ve
when taken +ve , solving the equation will give roots 1 and 2 , product will be 2
when taken -ve , solving equation will give 1 and 4 , which can be discarded as we have considered x and -ve and we are getting roots positive.
answer will be 2
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 11 Jan 2018, 10:34
I assume this is added by mistake into hard question. This is the easiest one.


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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]

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New post 19 Jan 2018, 11:19
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

Edited the answer choice C from -2 to 2 and the OA. It's C not E.



I just want to share my answer which could shorten calculation time a lot.

x^2 - 4x + 6 = 3 - |x - 1|
X^2 - 4x + 3 = - |x-1|
(X - 3) (x - 1) = - |x - 1|

If x = 1, both side = 0 => 1 is a root

If x not equal q, then:
(X-3) = - | x - 1| / (x-1)

If (x - 1) > 0, (x -3 ) = -1 => x = 2. [2 - 1]>0 => 2 is a root
If (x - 1) < 0, (x -3) = 1 => x = 4. [4-1] >0. => 4 is not a root


Conclusion: there are two root 1 and 2
Re: What is the product of all the solutions of x^2 - 4x + 6=3   [#permalink] 19 Jan 2018, 11:19

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