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# Is A positive? (1) x^2 - 2x + A is positive for all x (2)

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SVP
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Is A positive? (1) x^2 - 2x + A is positive for all x (2) [#permalink]

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26 Mar 2008, 06:01
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Is A positive?

(1) x^2 - 2x + A is positive for all x

(2) Ax^2 + 1 is positive for all x

Would someone please show me how to solve this? Also, what do they mean when they say "for all x"?
Manager
Joined: 02 Mar 2008
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26 Mar 2008, 06:05
(1) (x-1)^2 + A-1>0 for all x -> A>1
(2) Ax^2 + 1 > 0 for all x, consider A<0 -> 1-kx^2 ... can not >0 for all x

so D
SVP
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26 Mar 2008, 07:51
can you show me how you did that? what made you reach to such an equation? also, what does it mean exactly when it says "for all x"? i couldn't understand what it means exactly.
thanks
VP
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26 Mar 2008, 09:37
How is it D? I am between C & E cant A be -(1/2) and X be 1 in stat B?
Intern
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26 Mar 2008, 09:51
For choice B, suppose A was a negative fraction like -1/4?
Since X^2 is always positive. Let's say X = 1 or -1. Thus X^2 = 1.

[ (A) * (X^2) ]+1 =
[ (-1/4) * (1) ]+ 1 = 3/4, which is still positive. Since A is negative while answer is positive, A can be negative.

I'm for A.
Director
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26 Mar 2008, 10:18
tarek99 wrote:
Is A positive?

(1) x^2 - 2x + A is positive for all x

(2) Ax^2 + 1 is positive for all x

Would someone please show me how to solve this? Also, what do they mean when they say "for all x"?

Should be 'D'

Stmt1:
x^2 - 2x + A > 0
(x-1)^2 + (A -1) > 0

minimum value of (x-1)^2 is 0
in this case A > 1
Keep in mind that we have to find A so that eqn is valid for all x.
for (x-1)^2 >0, positive A very well satisfy the eqn. (x-1)^2 + (A -1) > 0
Hence for x^2 - 2x + A > 0, A will always be > 0
Suff

Stmt2:Ax^2 + 1 > 0
for all x, A should be +ve because for negative A, the equation may not hold good.
Suff

Hence 'D'
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27 Mar 2008, 01:05
the OA is D, but I don't understand how you guys ended up with such equations. can you please explain? also, what do they mean when they say "for all x"? didn't understand that part of the sentence.
Senior Manager
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27 Mar 2008, 02:42
"For all x" means for all the different types of values of x. The statement does not state that x is an integer (positive or negative)... so you have to assume what if x was positive, what if x was -ve. What if x was a fraction (+ve and -ve)....

Hope that makes sense
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27 Mar 2008, 07:35
I think the most general method is to consider for all possibility of x, normally by reducing the eqn become kind of (x-a)(x-b)... if possible. Another way is make use of those special cases such as x^2>=0 for all x...and GMAT may like to create a prob like the one posted here, which falls into kind of special eqn. That's why you see how we try to reduce it in eqn 1 to (x-1)^2.
Hard to tell the general rule, practicing more might help...
Re: Is A positive?   [#permalink] 27 Mar 2008, 07:35
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