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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.

Bunuel: Please explain the highlighted part. I didn't understand the real roots part.

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.

Bunuel: Please explain the highlighted part. I didn't understand the real roots part.

Re: Is A positive? x^2-2x+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

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.

Hi Bunuel, How did you infer that the parabola would be above X axis by looking at the equation?? Pls explain. Regards, H
_________________

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.

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 \(x^2-2x+A>0\) and told that this expression is positive for all x, which means that the parabola is above X-axis (otherwise it wouldn't be positive for all x).
_________________

We have \(x^2-2x+A>0\) and told that this expression is positive for all x, which means that the parabola is above X-axis (otherwise it wouldn't be positive for all x).

Thanks Bunuel, It was quite obvious.. dont know what was I thinking. Appreciate your help.
_________________

Re: Is A positive? x^2-2x+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^2-1??? also when ax^2+1 > or = 0 where is the third constant c??? is it 0???

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,\(X^2-2X+A > 0\) This 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, \(AX^2 + 1 > 0\) 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^2-2x+A>0 if we take the value 3 for example , 9-6+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^2-2x+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)^2-2(-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.
_________________

Re: Is A positive? x^2-2x+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^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. ....Want to know how +1 changes to -1 .....please explain

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. ....Want to know how +1 changes to -1 .....please explain

Re: Is A positive? x^2-2x+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?