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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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02 May 2013, 12:02
Thank u Bunuel. Good one about Max and Min Value of Quadratic Equations: http://www.youtube.com/user/patrickJMT?feature=watch



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Re: Is A positive? [#permalink]
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02 May 2013, 12:07
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 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. Bunuel: Please explain the highlighted part. I didn't understand the real roots part.



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Re: Is A positive? [#permalink]
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03 May 2013, 04:10
Rajkiranmareedu 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 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. Bunuel: Please explain the highlighted part. I didn't understand the real roots part. Check here: http://en.wikipedia.org/wiki/Quadratic_ ... scriminant
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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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03 May 2013, 13:45
Thanks for bearing with me The x^22x+A>0 is true for any value of X. Hence, there is no distinct roots for the equation so, Discriminant is less than zero.



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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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03 May 2013, 16:33
I actually did the +1,1 trick and got the answer but i would also like to know your train of thought, when you use the concept of the parabola. May be when you see Quadratic equation, you think about descrimants,parabola..every possible angle



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Re: Is A positive? [#permalink]
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19 May 2013, 00:51
Bunuel wrote: 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.
Hi Bunuel, How did you infer that the parabola would be above X axis by looking at the equation?? Pls explain. Regards, H
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Re: Is A positive? [#permalink]
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19 May 2013, 04:30
imhimanshu wrote: Bunuel wrote: 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.
Hi Bunuel, How did you infer that the parabola would be above X axis by looking at the equation?? Pls explain. Regards, H We have \(x^22x+A>0\) and told that this expression is positive for all x, which means that the parabola is above Xaxis (otherwise it wouldn't be positive for all x).
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Re: Is A positive? [#permalink]
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19 May 2013, 08:45
Bunuel wrote: We have \(x^22x+A>0\) and told that this expression is positive for all x, which means that the parabola is above Xaxis (otherwise it wouldn't be positive for all x).
Thanks Bunuel, It was quite obvious.. dont know what was I thinking. Appreciate your help.
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Re: Is A positive? [#permalink]
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16 Sep 2013, 07:08
Bunuel wrote: So what is the least value of \(x^22x\)? The least value of quadratic expression \(ax^2+bx+c\) is when \(x=\frac{b}{2a}\)
Don't know much about this rule, about the least value of a quadratic. How did we get this? Is there a way to prove this. Thank you
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Re: Is A positive? [#permalink]
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Re: Is A positive? [#permalink]
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16 Sep 2013, 07:33
Bunuel wrote: stne wrote: Bunuel wrote: So what is the least value of \(x^22x\)? The least value of quadratic expression \(ax^2+bx+c\) is when \(x=\frac{b}{2a}\)
Don't know much about this rule, about the least value of a quadratic. How did we get this? Is there a way to prove this. Thank you Yes. Check the last chapter here: mathcoordinategeometry87652.htmlHope it helps. Thank You +1 that definitely helps.
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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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17 Sep 2013, 11:28
is A^2 + 1 quadratic??



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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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18 Sep 2013, 00:19



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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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18 Sep 2013, 10:33
Bunuel wrote: skamran wrote: is A^2 + 1 quadratic?? Do you mean Ax^2 + 1? There is no A^2 + 1 in the question... yeh i meant ax^2+ 1, i know how to solve quadratic equations,also i know in the question we have been told that ax^2+1 is positive for all xes, what could be the case if the equation was ax^21??? also when ax^2+1 > or = 0 where is the third constant c??? is it 0???



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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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18 Sep 2013, 10:40
i guess its the form of (a+b)^2???...



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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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24 Sep 2013, 02:18
How do we derive the formula for calculating the vertex of a parabola?



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Re: DS: number properties [#permalink]
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24 Sep 2013, 20:44
VeritasPrepKarishma wrote: 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 Responding to a pm: Quote: I still did not understand your solution x^22x+A>0 if we take the value 3 for example , 96+A>0 3+A>0 which gives A>()3 so A can assume 2,1,0 and so on and we still get the overall value as +ve. Can you help me understand what i am missing ? Given that x^22x+A is always positive. No matter what the value of x, the value of A is such that this expression is always positive. Whether x = ...2, 0, 1, 4, 100..., the expression will always be positive. So let's put a few values of x. Put x = 2 (2)^22(2)+A > 0 A > 8 Put x = 0 0^2  2*0 + A > 0 A > 0 Put x = 1 1^2  2*1 + A > 0 A > 1 Put x = 3 3^2  2*3 + A > 0 A > 3 and so on... So we see that A must be greater than 8, it should also be greater than 3, it should also be greater than 0 and it should also be greater than 1. So what values do you think A can take? Values which are greater than all these values i.e. values like 8, 10 etc. In any case, we are asked whether A is positive and we know that it must be greater than 1. Hence, we know that A must be positive. Sufficient.
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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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17 Oct 2013, 04:46
Dear Brunel
Please explain last line ....
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. ....Want to know how +1 changes to 1 .....please explain



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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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17 Oct 2013, 04:56



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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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22 Oct 2013, 15:00
there are many hard data sufficiency questions? To get over 700 in GMAT at least how many hard data sufficiency questions do we have to answer? I have a lot of problems with hard and tricky DS questions. I always go close to the answer but finally make mistake in hard DS by not noticing one or two things. Can anyone help me please?




Re: Is A positive? x^22x+A is positive for all x Ax^2+1 is
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