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# Is A positive? x^2-2x+A is positive for all x Ax^2+1 is

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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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02 May 2013, 11:02
Thank u Bunuel.

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02 May 2013, 11:07
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

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.

Bunuel: Please explain the highlighted part. I didn't understand the real roots part.
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03 May 2013, 03:10
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Rajkiranmareedu 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

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.

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

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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, 12:45
Thanks for bearing with me

The x^2-2x+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^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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03 May 2013, 15:33
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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18 May 2013, 23:51
Bunuel wrote:
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
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19 May 2013, 03:30
imhimanshu wrote:
Bunuel wrote:
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).
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19 May 2013, 07:45
Bunuel wrote:

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.
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16 Sep 2013, 06:08
Bunuel wrote:
So what is the least value of $$x^2-2x$$? The least value of quadratic expression $$ax^2+bx+c$$ is when $$x=-\frac{b}{2a}$$

How did we get this? Is there a way to prove this.
Thank you
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16 Sep 2013, 06:12
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stne wrote:
Bunuel wrote:
So what is the least value of $$x^2-2x$$? The least value of quadratic expression $$ax^2+bx+c$$ is when $$x=-\frac{b}{2a}$$

How did we get this? Is there a way to prove this.
Thank you

Yes. Check the last chapter here: math-coordinate-geometry-87652.html

Hope it helps.
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16 Sep 2013, 06:33
Bunuel wrote:
stne wrote:
Bunuel wrote:
So what is the least value of $$x^2-2x$$? The least value of quadratic expression $$ax^2+bx+c$$ is when $$x=-\frac{b}{2a}$$

How did we get this? Is there a way to prove this.
Thank you

Yes. Check the last chapter here: math-coordinate-geometry-87652.html

Hope it helps.

Thank You +1 that definitely helps.
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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17 Sep 2013, 10:28
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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17 Sep 2013, 23:19
skamran wrote:

Do you mean Ax^2 + 1? There is no A^2 + 1 in the question...
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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, 09:33
Bunuel wrote:
skamran wrote:

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???
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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, 09:40
i guess its the form of (a+b)^2???...
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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

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24 Sep 2013, 01:18
How do we derive the formula for calculating the vertex of a parabola?
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24 Sep 2013, 19:44
VeritasPrepKarishma wrote:
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,$$X^2-2X+A > 0$$
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.

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.
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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, 03:46
Dear Brunel

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
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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, 03:56
archit wrote:
Dear Brunel

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

You mean to know how we get A>1 from -1+A>0?

Add 1 to both sides of -1+A>0 --> A>1.

Hope it's clear.
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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, 14:00
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^2-2x+A is positive for all x Ax^2+1 is   [#permalink] 22 Oct 2013, 14:00

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