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Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]
 15 Jul 2010, 12:42
Question Stats:
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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
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Last edited by Bunuel on 11 Apr 2012, 00:06, edited 1 time in total.
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Re: Is A positive? [#permalink]
15 Jul 2010, 14:40
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noboru wrote: Is A positive?
x^2-2x+A is positive for all x
Ax^2+1 is positive for all x
OA is A Is A>0? (1) x^2-2x+A is positive for all x: Quadratic expression x^2-2x+A is a function of of upward parabola (it's upward as coefficient of x^2 is positive). We are told that this expression is positive for all x --> x^2-2x+A>0, which means that this parabola is "above" X-axis OR in other words parabola has no intersections with X-axis OR equation x^2-2x+A=0 has no real roots. Quadratic equation to has no real roots discriminant must be negative --> D=2^2-4A=4-4A<0 --> 1-A<0 --> A>1. Sufficient. (2) Ax^2+1 is positive for all x: Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too. Not sufficient. Answer: A.
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Re: Is A positive? [#permalink]
16 Jul 2010, 05:21
Hi,
I dont get it sorry... I mean I understand your equations Bunuel, but I tried first with picking numbers:
If I pick -0.5 for x --> x^2-2x+A>0 will hold for A > -1.25
...
Where is my mistake??
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Re: Is A positive? [#permalink]
16 Jul 2010, 08:01
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AndreG wrote: Hi,
I dont get it sorry... I mean I understand your equations Bunuel, but I tried first with picking numbers:
If I pick -0.5 for x --> x^2-2x+A>0 will hold for A > -1.25
...
Where is my mistake?? The point here is that x^2-2x+A>0 for all x-es. Let's do this in another way: We have (x^2-2x)+A>0 for all x-es. The sum of 2 quantities ( x^2-2x and A) is positive for all x-es. So for the least value of x^2-2x, A must make the whole expression positive. So what is the least value of x^2-2x? The least value of quadratic expression ax^2+bx+c is when x=-\frac{b}{2a}, so in our case the least value of x^2-2x is when x=-\frac{-2}{2}=1 --> x^2-2x=-1 --> -1+A>0 --> A>1. OR: x^2-2x+A>0 --> x^2-2x+1+A-1>0 --> (x-1)^2+A-1>0 --> least value of (x-1)^2 is zero thus A-1 must be positive ( 0+A-1>0)--> A-1>0 --> A>1. Hope it's clear.
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Re: Is A positive? [#permalink]
16 Jul 2010, 08:26
Wow u rock man!  That was very clear! I especially like the +1 -1 trick Posted from my mobile device 
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Re: Is A positive? [#permalink]
16 Jul 2010, 09:36
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Hi Bunuel,
I really liked approached here but I still have some confusion,
Say for e.g if try to pick the numbers say x = -3
Then the equation in the first statement becomes
x^2 - 2x + A = 9 +6 +A = 15 + A >0
So now if we see A can have -ve and +ve values, isnt it ?
I am confused with this.
Please explain, whats wrong with this one.
Cheers
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Re: Is A positive? [#permalink]
16 Jul 2010, 09:49
nitishmahajan wrote: Hi Bunuel,
I really liked approached here but I still have some confusion,
Say for e.g if try to pick the numbers say x = -3
Then the equation in the first statement becomes
x^2 - 2x + A = 9 +6 +A = 15 + A >0
So now if we see A can have -ve and +ve values, isnt it ?
I am confused with this.
Please explain, whats wrong with this one.
Cheers Not every question can be solved by number picking. For all x-es means that no matter what x you pick x^2 - 2x + A must be positive. So it must be positive even for the lowest value of x^2 - 2x which is -1 --> so -1+A must be positive hence A must be more than 1. Now again: if A>1 then for any x expression x^2 - 2x + A is positive. But if A=-15 (or any other number less than 1) we can find some x-es for which expression x^2 - 2x + A is not positive, so theese values of A (values of A\leq{1}) are not valid. Hope it's clear.
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Re: Is A positive? [#permalink]
16 Jul 2010, 09:56
Bunuel wrote: nitishmahajan wrote: Hi Bunuel,
I really liked approached here but I still have some confusion,
Say for e.g if try to pick the numbers say x = -3
Then the equation in the first statement becomes
x^2 - 2x + A = 9 +6 +A = 15 + A >0
So now if we see A can have -ve and +ve values, isnt it ?
I am confused with this.
Please explain, whats wrong with this one.
Cheers Not every question can be solved by number picking. For all x-es means that no matter what x you pick x^2 - 2x + A must be positive. So it must be positive even for the lowest value of x^2 - 2x which is -1 --> so -1+A must be positive hence A must be more than 1. Now again: if A>1 then for any x expression x^2 - 2x + A is positive. But if A=-15 (or any other number less than 1) we can find some x-es for which expression x^2 - 2x + A is not positive, so theese values of A (values of A\leq{1}) are not valid. Hope it's clear. Thanks for the reply Bunuel, I understood the approach but the fact which is baffling me is that say the equation after subsituting value of x=-3 i.e 15+ A > 0 now we can have a value of A=-3 or may be -4 etc and still have the value of the equation in statement 1 as +ve Am I thinking too much or just lacking some thing basic concept. I appreciate your patience.
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Re: Is A positive? [#permalink]
16 Jul 2010, 10:16
nitishmahajan wrote: Bunuel wrote: nitishmahajan wrote: Hi Bunuel,
I really liked approached here but I still have some confusion,
Say for e.g if try to pick the numbers say x = -3
Then the equation in the first statement becomes
x^2 - 2x + A = 9 +6 +A = 15 + A >0
So now if we see A can have -ve and +ve values, isnt it ?
I am confused with this.
Please explain, whats wrong with this one.
Cheers Not every question can be solved by number picking. For all x-es means that no matter what x you pick x^2 - 2x + A must be positive. So it must be positive even for the lowest value of x^2 - 2x which is -1 --> so -1+A must be positive hence A must be more than 1. Now again: if A>1 then for any x expression x^2 - 2x + A is positive. But if A=-15 (or any other number less than 1) we can find some x-es for which expression x^2 - 2x + A is not positive, so theese values of A (values of A\leq{1}) are not valid. Hope it's clear. Thanks for the reply Bunuel, I understood the approach but the fact which is baffling me is that say the equation after subsituting value of x=-3 i.e 15+ A > 0 now we can have a value of A=-3 or may be -4 etc and still have the value of the equation in statement 1 as +ve Am I thinking too much or just lacking some thing basic concept. I appreciate your patience. I think you just don't understand one thing in statement (1): x^2-2x+A>0 FOR ALL x-es. You say that if x=-3 then A can be for example -10 (or any number more than -15) and x^2-2x+A will be positive, but if x=1 does A=-10 makes x^2-2x+A positive? NO! So you should find such value of A (such range) for which x^2-2x+A is positive no matter what value of x you'll plug. And the way how to find this range is shown in my previous posts.
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Re: Is A positive? [#permalink]
16 Jul 2010, 10:23
Thanks Bunuel, Now I understood, I appreciate your patience in making me understand this one ..! Cheers,
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Re: Is A positive? [#permalink]
27 Jul 2010, 15:47
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And all this has to come to me in less than 2 mins?
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DS: number properties [#permalink]
10 Apr 2012, 19:17
Is A positive? 1) X^2-2X+A is positive for all X 2) AX^2 + 1 is positive for all X given answer as A...but i thought it should be E.. source: hard problems from gmatclub tests number properties I
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Re: DS: number properties [#permalink]
10 Apr 2012, 21:03
1) X^2-2X+A is positive for all X I think A could not be the answer, for example, if A = 0, and X = 4, then also the expression is positive, but A = 0 is neither positive nor negative
Again, if A = 1, and and X = 4, then also the expression is positive2) AX^2 + 1 is positive for all X Same logic as above, if A is 0, then the expression is positive, and the expression is also postive for any value of X where A > 0
In a nutshell, I too think the answer is E.
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Re: DS: number properties [#permalink]
11 Apr 2012, 10:42
rongali wrote: Is A positive? 1) X^2-2X+A is positive for all X 2) AX^2 + 1 is positive for all X given answer as A...but i thought it should be E.. source: hard problems from gmatclub tests number properties I 1) X^2-2X+A is positive for all X For all values of X, X^2-2X+A > 0This means, for X = 0, X^2-2X+A > 0; for X = 1, X^2-2X+A > 0; for X = -2, X^2-2X+A > 0 etc etc etc Let's put X = 0. 0^2-2*0+A > 0 should hold. Therefore, A > 0 should hold. Sufficient. 2) AX^2 + 1 is positive for all X For all X, AX^2 + 1 > 0Here, A could be positive or A could be 0 (since, when A = 0, we get 1 > 0 which holds no matter what the value of X.) Since A can be 0, we cannot say whether A is positive. Not Sufficient. Answer A
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Re: Is A positive? [#permalink]
04 Jul 2012, 03:55
Bunuel wrote: noboru wrote: Is A positive?
x^2-2x+A is positive for all x
Ax^2+1 is positive for all x
OA is A (2) Ax^2+1 is positive for all x: Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too. Not sufficient. Answer: A. Why didn't you use the discriminant formula to assess statement 2? I tried the discriminant rule and got a>0. I had 0-4a<0 which turns to a>0. What am I missing here? Thanks, Diana
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Re: Is A positive? [#permalink]
04 Jul 2012, 03:59
dianamao wrote: Bunuel wrote: noboru wrote: Is A positive?
x^2-2x+A is positive for all x
Ax^2+1 is positive for all x
OA is A (2) Ax^2+1 is positive for all x: Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too. Not sufficient. Answer: A. Why didn't you use the discriminant formula to assess statement 2? I tried the discriminant rule and got a>0. I had 0-4a<0 which turns to a>0. What am I missing here? Thanks, Diana You are right: if we use the same approach for (2) then we'll get A>0 BUT if A=0 then Ax^2+1 won't be a quadratic function anymore. So this approach will work only if A doesn't equal to zero, but we can not eliminate this case and if A=0 then Ax^2+1=1 is also always positive. Hence Ax^2+1 is positive for A>0 (if we use quadratic function approach) as well as for A=0, so for A\geq0. Hope it's clear.
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This is a question from gmat club tests. Is A positive? 1. x^2 -2x +A is positive for all x. 2. A*x^2 +1 is positive for all x. I got E and my way of solving is as below: St1. x^2 - 2x +A > 0 Let x=0, so A>0. Let x=-1, so A>-3. In this case A can be negative or positive. Insufficient. St2. A*x^2 +1 > 0 Let x=-1, so A>-1. Again A can be positive or negative. Insufficient. St1+St2: Let x=-1, so A > -1. Again A can be positive or negative. Insufficient. So it's E. However the OA is not E. Please advise.
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Re: Is A positive? [#permalink]
20 Aug 2012, 01:36
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]
21 Aug 2012, 16:51
best way to deal this problem is to bet on A more than X.. it wud b yes if A>0 Or No ,if A<0 .... then first assume A>0 , then check whether statement 1 & 2 is true or not for all value of X.... then assume A<0 ,then check whether statement 1 & 2 is true or not for all value of X....
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Re: Is A positive? [#permalink]
21 Aug 2012, 19:57
dianamao wrote: Bunuel wrote: noboru wrote: Is A positive?
x^2-2x+A is positive for all x
Ax^2+1 is positive for all x
OA is A (2) Ax^2+1 is positive for all x: Ax^2+1>0 --> when A\geq0 this expression is positive for all x. So A can be zero too. Not sufficient. Answer: A. Why didn't you use the discriminant formula to assess statement 2? I tried the discriminant rule and got a>0. I had 0-4a<0 which turns to a>0. What am I missing here? Thanks, Diana @Diana - Which discriminant rule did you use?
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Re: Is A positive?
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21 Aug 2012, 19:57
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