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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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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, 18:03
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
1
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
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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 Updated on: 20 Jun 2018, 13:54
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.



pushpitkc , Bunuel, @niks18
can you please explain, looking at Bunuel`s solution in which case when testing values do i need use more or equal sign VS just more / less sign

\(x<1\) VS \(x\geq{1}\) and why in case TWO we dont use this \(x>1\) :? instead of \(x\geq{1}\)

or why in case one \(x<1\) we exclude "less or equal" :? unlike case 2 :?

on the other hand in the post about important properties both signs (more than / less than) are combined with "equal" sign :?

https://gmatclub.com/forum/absolute-val ... 75002.html
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Originally posted by dave13 on 20 Jun 2018, 11:51.
Last edited by dave13 on 20 Jun 2018, 13:54, edited 1 time in total.
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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 20 Jun 2018, 13:53
1
dave13 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.

Answer: C.



pushpitkc , Bunuel
can you please explain, looking at Bunuel`s solution in which case when testing values do i need use more or equal sign VS just more / less sign

\(x<1\) VS \(x\geq{1}\) and why in case TWO we dont use this \(x>1\) :? instead of \(x\geq{1}\)


on the other hand in the post about important properties both signs (more than / less than) are combined with "equal" sign :?

https://gmatclub.com/forum/absolute-val ... 75002.html


Hey dave13

Bunuel has already clarified your doubt here
https://gmatclub.com/forum/what-is-the- ... l#p1183663

The basic idea is that you can have both the possibilities \(x>1\) and \(x \leq 1\) or \(x \geq1\) and \(x<1\)

Hope this helps you!
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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 21 Jun 2018, 07:15
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.




hello generis :-) can you please explain how Bunuel got this \(x<1\), I think I am doing something wrong ,

if Negative I rewrite it as \(|x - 1|<0\) so I get \(-(x-1)< 0\), ---> \(-x+1<0\) --> \(-x<-1\) ---> divide each side by \(-1\) and get I \(x>1\) :?

same issue with positive case...

i followed your explanation here :-) https://gmatclub.com/forum/what-is-the- ... s#p2081311

"When
(x+2) is positive (one of two "original conditions")
x+2>0
x>−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 21 Jun 2018, 18:45
1
dave13 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.

Answer: C.

hello generis :-) can you please explain how Bunuel got this \(x<1\), I think I am doing something wrong ,

if Negative I rewrite it as (1)\(|x - 1|<0\)

(2) I get \(-(x-1)\)
<0
, ---> \(-x+1<0\) --> \(-x<-1\) ---> divide each side by \(-1\) and get I \(x>1\) :?

same issue with positive case... i followed your explanation here :-) https://gmatclub.com/forum/what-is-the- ... s#p2081311

"When (x+2) is positive (one of two "original conditions")
x+2>0
x>−2


dave13 , you are really close. You made two small missteps, easily corrected.
The stricken part is not correct, but I can understand why you did what you did. There is some confusion because ...

We actually have 3 different but related things going on:

--- specifying the negative case: \((x-1)<0\)

--- defining a condition, an allowable range for the solutions: (\(x>1\))

--- in simplest form, writing what we will actually plug into the equation: \((1-x)\)

Quote:
how Bunuel got this \(x<1\)

He got that inequality from "If (x-1) < 0 ..." Work backward from what he wrote
\(x<1\)
Subtract 1 from both sides:
\(x-1<(1-1)\), thus:
\(x-1<0\)

In other words, when Bunuel writes \(x<1\), that statement is derived from \(x-1<0\) (the "negative" case). He adds 1 to both sides to get \(x<1\)

That \(x<1\) is sort of SEPARATE. \(x<1\) gives us a range for allowed solutions. Write that condition and ignore it until we have an answer.

You wrote,
Quote:
if Negative I rewrite it as \(|x - 1|<0\)

Almost. Take the absolute value bars off and replace with parentheses, thus \((x-1)<0\)

We are really evaluating what is inside the bars.

What is inside? The expression x - 1
So we write (x-1), not |x-1|
In more familiar lingo, perhaps, we "remove the [absolute value] brackets."

AND: We cannot write |x-1| = -1
Absolute value cannot be negative because it is a distance from 0, and distances are not negative. The expression inside the absolute bars can be negative.

You wrote:
Quote:
\(-(x-1)< 0\)

Almost. Separate it into two parts
\((x-1)<0\): "I am considering the case in which the expression is negative"

\(-(x-1)\): "After I rewrite it a little, I will insert THIS term into the equation."

Steps, Part I:

Step 1) write down that you are considering the negative case, i.e., \((x-1)<0\).

Step 2) separately, decipher what we will actually insert into the equation
--put the negative sign in front of the expression, just as you did: \(-(x-1)\)

Put the expression in simplest terms
\(-(x-1)=\)
\((-x+1)\). Now switch the numbers' order and get
\((1-x)\)
THAT term goes into the equation

Step 3) Before we put that term into the equation, find the condition.
From \((x-1)<0\):
\(x-1+1<0+1\)
\(x<1\)
We ignore that term for a bit

Step 4) Plug #2 -- the whole thing-- into the equation
\(x^2 - 4x + 6 = 3 - |x - 1|\)
\(x^2 - 4x + 6 = 3 - (1-x)\)
\(x^2 - 4x + 6 = 3 - 1+x\)
\(x^2 - 5x + 4 = 0\)

Step 5) Factor to get the roots and solve:
\((x-1)(x-4)=0\)
\((x-1)=0\), so \(x=1\) OR
\((x-4)=0\), so \(x=4\)

Step 6: Check those answers against the "condition," the range we allow from Step 3.
Do the solutions 1 and 4 fall in the allowed range of \(x<1\)? No. 1 and 4 are not smaller than 1. Those solutions are not valid.

Part II -- For the non-negative case (the EXPRESSION is non-negative)

1) Write "x is greater than or equal to 0" in mathematical terms

2) write, separately, what you will actually put into the equation. Put a positive sign in front of the expression.(The "result" when we put a negative sign changed the inside. This one shouldn't.)

3) Write, separately, the "condition" part (get \(x\) on one side). Ignore till the end

4) Plug #2 into the equation

5) Get everything on LHS, RHS = 0

6) factor the quadratic and solve (set each factor equal to 0)

7) check your solutions against the allowed range for THIS PART's Step 3. Are those solutions in your allowed range from #3?

Last step: if the solutions are valid, do what the prompt asks.

As mentioned, we are doing three different but related things...

\((x-1)<0\) announces: I am considering the negative case

\(x<1\) says: here is the allowable range for my solution, and

\(-(x-1)=\)
\((1-x)\) is what actually gets plugged into the equation.

Let me know how it goes. I hope that 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 09 Aug 2018, 05:03
Dear Bunuel,

Just not clear of one point, why shouldn't it be x>= -2 (when x+2>=0), |x+2|=x+2; instead of what you have written without '=' sign: x>-2 (when x+2>0), |x+2|=x+2
Re: What is the product of all the solutions of x^2 - 4x + 6=3 &nbs [#permalink] 09 Aug 2018, 05:03

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