djmobley wrote:
Statement I is suff, huh? I don't want to sound like I'm degrading anyone's fancy math - but there seems to be a serious disconnect happening here.
I - x^2-2X+A has to be positive for ALL xs. Okay - so all I have to do is find at least 1 x value where A can be either positive or negative.
x=-1 ---> 1+2+A=positive ---> 3+A=positive. My math may not be as fancy as everyone else's here but I'm pretty sure A needs to be greater than negative 3 with this X value thus A has the potential to be a positive or negative value.
A should not be suff...?
I believe this is a GMAT Club question created by Bunnel. You can literally take a semester's worth of a course to study the theory on a single one of the questions Bunnel creates. I don't think most people can gasp the genius thought on some of these questions created by Bunnel, IanStewart and a few others.
Obviously the confusion here is what the heck "
all x" means.
This question by no means is wrong. It is however very poorly written at least as far as GMAT standards are concerned. The question is very confusing and not even remotely in the ballpark of what I'd consider fair game for the GMAT.
WRONG If you are getting Statement (1) wrong it is because you are assuming "all x" means for any specific given value of x. If you do that then you'll start picking numbers to plug into the equation x^2 - 2x + a > 0
For example if x = 5, then
x^2 - 2x + a > 0
(5)^2 - 2(5) + a > 0
25-10+a>0
15+a>0
a>-15
Here you can see a>-15 and say a can be -14 or a can be 30 and say insufficient. But, this is wrong because "all x" does not mean any specific value of x.
CORRECT"all x" means EVERY single possible value of x. This basically means x is EVERY number in the universe. X is 5, X is 0, X is the amount of money in my checking account (-56.20)
, X is be the weight of the earth (5.972 × 10^24). The question means for EVERY value of x, x^2 - 2x + a > 0.
If you simply the right part of the equation you get:
x(x-2) + a > 0
To simply it even more, we can assign a value of "b" to the left part of the equation, so x(x-2) = b. So now we have b + a > 0
Again, the question asks for EVERY value of x, is a in the above equation positive?
Let's try WHEN b =0 (noticed I said WHEN b = 0, not "if" b=0. There isn't an if b=0, it is when b=0)
so for b= x(x-2) =0, then x = -2 or 0. plug x=0 into x(x-2) + a > 0. 0 + a > 0, so
a > 0.
Let's try WHEN b = negative number, so b= x(x-2) < 0, then x is between 0 and 2, so let's try x = 1 into the equation x(x-2) + a > 0. -1 + a>0, so
a>1.Let's try WHEN b = positive number, so b= x(x-2) > 0, then x is less than 0 -or- x is more than 2. Let's try x=3 into the equation x(x-2) + a > 0. So 3+a>0, then
a > -3.
So, the solution set so far is a>0, a>1, a>-3, we can try different numbers to get more of a solution set, but there is no need at this point, since the most restrictive of the solution set is a>1, so we know a is at least >1, so a is positive.