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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
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]

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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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Problem Solving Absolute Value [#permalink]

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New post 03 Nov 2014, 17:02
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]

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New post 04 Nov 2014, 03:49
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.

Also, please read carefully and follow: rules-for-posting-please-read-this-before-posting-133935.html Pay attention to rules 1, 3 and 5. Thank you.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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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
Expert's post
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] 11 Aug 2015, 18:15

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