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

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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

Originally posted by noboru on 15 Jul 2010, 12:42.
Last edited by Bunuel on 12 Apr 2018, 01:57, edited 2 times in total.
Edited the question
Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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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.

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

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2
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??
Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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19
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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? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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Wow u rock man! That was very clear!

I especially like the +1 -1 trick

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Manager  Joined: 25 Jun 2009
Posts: 225
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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1
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
Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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

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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.

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. Math Expert V
Joined: 02 Sep 2009
Posts: 59725
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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1
1
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.

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? 1. x^2 - 2x + A is +ve for all x 2. (A*x^2) +  [#permalink]

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asmit123 wrote:
Is A positive?

1. x^2 - 2x + A is +ve for all x
2. (A*x^2) + 1 is +ve for all x

I've posted about this question a couple of times. Many people solve this backwards and arrive at the wrong answer (those test takers relying on 'number picking' strategies almost always answer this question incorrectly). The important word in each of the two statements is 'all'. In Statement 1, x^2 - 2x + A is positive not just for some value of x; it must be positive for EVERY value of x. In particular, it's positive when x=0, so substituting x=0, we learn instantly that A is positive and Statement 1 is sufficient.

Statement 2 is also almost sufficient. It is only insufficient because of a technicality: it's possible that A=0.

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

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4
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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.

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

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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.

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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Posts: 59725
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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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.

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

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

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

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

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

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

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

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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? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1   [#permalink] 22 Oct 2013, 15:00

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