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Is A positive? x^22x+A is positive for all x Ax^2+1 is [#permalink]
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15 Jul 2010, 12:42
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Is A positive? (1) x^22x+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]
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noboru wrote: Is A positive?
x^22x+A is positive for all x Ax^2+1 is positive for all x
OA is A Is \(A>0\)? (1) \(x^22x+A\) is positive for all \(x\): Quadratic expression \(x^22x+A\) 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 \(x\) > \(x^22x+A>0\), which means that this parabola is "above" Xaxis OR in other words parabola has no intersections with Xaxis OR equation \(x^22x+A=0\) has no real roots. Quadratic equation to has no real roots discriminant must be negative > \(D=2^24A=44A<0\) > \(1A<0\) > \(A>1\). 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]
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16 Jul 2010, 05:21
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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^22x+A>0 will hold for A > 1.25
...
Where is my mistake??



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Re: Is A positive? [#permalink]
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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^22x+A>0 will hold for A > 1.25
...
Where is my mistake?? The point here is that \(x^22x+A>0\) for all \(xes\). Let's do this in another way: We have \((x^22x)+A>0\) for all \(xes\). The sum of 2 quantities (\(x^22x\) and \(A\)) is positive for all \(xes\). So for the least value of \(x^22x\), \(A\) must make the whole expression positive. So what is the least value of \(x^22x\)? The least value of quadratic expression \(ax^2+bx+c\) is when \(x=\frac{b}{2a}\), so in our case the least value of \(x^22x\) is when \(x=\frac{2}{2}=1\) > \(x^22x=1\) > \(1+A>0\) > \(A>1\). OR: \(x^22x+A>0\) > \(x^22x+1+A1>0\) > \((x1)^2+A1>0\) > least value of \((x1)^2\) is zero thus \(A1\) must be positive (\(0+A1>0\))> \(A1>0\) > \(A>1\). Hope it's clear.
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Re: Is A positive? [#permalink]
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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]
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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]
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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 \(xes\) 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 \(xes\) 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]
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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 \(xes\) 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 \(xes\) 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]
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16 Jul 2010, 10:16
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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 \(xes\) 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 \(xes\) 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): \(x^22x+A>0\) FOR ALL \(xes\). You say that if \(x=3\) then \(A\) can be for example 10 (or any number more than 15) and \(x^22x+A\) will be positive, \(but\) if \(x=1\) does \(A=10\) makes \(x^22x+A\) positive? NO! So you should find such value of \(A\) (such range) for which \(x^22x+A\) 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]
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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]
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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]
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10 Apr 2012, 19:17
Is A positive? 1) X^22X+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]
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10 Apr 2012, 21:03
1) X^22X+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]
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11 Apr 2012, 10:42
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rongali wrote: Is A positive? 1) X^22X+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^22X+A is positive for all X For all values of X,\(X^22X+A > 0\) This means, for X = 0, \(X^22X+A > 0\); for X = 1, \(X^22X+A > 0\); for X = 2, \(X^22X+A > 0\) etc etc etc Let's put X = 0. \(0^22*0+A > 0\) should hold. Therefore, A > 0 should hold. Sufficient. 2) AX^2 + 1 is positive for all X For all X, \(AX^2 + 1 > 0\) Here, 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]
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04 Jul 2012, 03:55
Bunuel wrote: noboru wrote: Is A positive?
x^22x+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 04a<0 which turns to a>0. What am I missing here? Thanks, Diana



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Re: Is A positive? [#permalink]
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04 Jul 2012, 03:59
dianamao wrote: Bunuel wrote: noboru wrote: Is A positive?
x^22x+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 04a<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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Is A positive? [#permalink]
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19 Aug 2012, 19:11
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]
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Re: Is A positive? x^22x+A is positive for all x Ax^2+1 is [#permalink]
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21 Aug 2012, 16:51
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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]
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21 Aug 2012, 19:57
dianamao wrote: Bunuel wrote: noboru wrote: Is A positive?
x^22x+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 04a<0 which turns to a>0. What am I missing here? Thanks, Diana @Diana  Which discriminant rule did you use?
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