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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]  06 Jul 2013, 08:53
Expert's post
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]  09 Jul 2013, 14:25
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]  19 Jun 2014, 01:57
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.

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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Problem Solving Absolute Value [#permalink]  03 Nov 2014, 16:02
What is the product of all the solutions of (x^2) - (4x) + (6)= lx - 1l ?
a)-8
b)-4
c)-2
d)4
e)8

*Source: Total GMAT Math

My follow-up question is that since the absolute value will always be a positive number, can you automatically eliminate answer choices a, b, and c?

Thanks!
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Re: What is the product of all the solutions of x^2 - 4x + 6=3 [#permalink]  04 Nov 2014, 02:49
Expert's post
ferlytate wrote:
What is the product of all the solutions of (x^2) - (4x) + (6)= lx - 1l ?
a)-8
b)-4
c)-2
d)4
e)8

*Source: Total GMAT Math

My follow-up question is that since the absolute value will always be a positive number, can you automatically eliminate answer choices a, b, and c?

Thanks!

Merging similar topics. Please refer to the discussion on page 1.

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Re: What is the product of all the solutions of x^2 - 4x + 6=3   [#permalink] 04 Nov 2014, 02:49

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