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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] New post 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] New post 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] New post 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.

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
Re: What is the product of all the solutions of x^2 - 4x + 6=3   [#permalink] 19 Jun 2014, 01:57
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