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

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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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Updated on: 12 Apr 2018, 01:57
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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
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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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15 Jul 2010, 14:40
21
19
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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16 Jul 2010, 08:01
21
7
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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11 Apr 2012, 10:42
4
3
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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24 Sep 2013, 20:44
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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16 Jul 2010, 09:36
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
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Joined: 02 Sep 2009
Posts: 60605
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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16 Jul 2010, 10:16
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 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 positive for all x (2) Ax^2 + 1  [#permalink]

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22 Oct 2013, 23:39
1
bsahil wrote:
VeritasPrepKarishma wrote:

X = 10 is fine but it doesn't help. We know that this inequality holds for all x. We need to plug in a value for x which tells us something about A. If we put x = 0, we are left with just A and that will tell us something about A. Just plugging in any value may not work; you have to look for a smart value.

So for this question, no other value of x(except x=0,2) gives us any information about A. hence we take only that value which gives us explicit and fixed result for A. Am I correct??
Similarly we would do in other similar questions as well...??

No, look, we know that $$x^2 - 2x + A > 0$$ for all x. For every value of x, this inequality should be satisfied.

Put x = 0, you get A > 0
Put x = 1, you get $$1 - 2 + A > 0 i.e. A > 1$$
Put x = 10, you get A > -80

Now the point is that A should take a value such that all these conditions are satisfied. Say A can be 5. If A is 5, it is > 0, > 1 and > -80.

When I look at $$x^2 - 2x + A > 0$$ given x can take any value, the first value that pops in my head to get a sense of A is x = 0. That provides exactly what I need. Had the question been whether A > 2, x = 0 would not have helped. I would have had to search a little for a pattern to see how the value of x changes. This question is made in a way that x = 0 helps immediately. Anyway, it is a good idea to try the value 0 in many circumstances. It simplifies things immensely and usually helps you eliminate a couple of options at least.
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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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23 Oct 2013, 21:56
1
ratanpandit wrote:
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.

if we put x=-3 in 1, then A can have -1 or -2 value also ??

No, A can never have a value of -1 or -2. It must be greater than 1.
If we put x = -3, we get
X^2-2X+A > 0
A > -15
So this tells us that A must be greater than -15. Putting other values of x such as 0, 1, 2 etc tell us that A must be greater than 0 and A must be greater than 1 etc. Since this inequality holds for ALL values of x, A must be greater than 1 because a value greater than 1 will automatically be greater than -15 as well as 0. If we take a value of A such as -14, it will be greater than -15 but not greater than 0 or 1 hence the inequality will not hold 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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16 Jul 2010, 05:21
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??
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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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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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Posts: 60605
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#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 $$x-es$$ 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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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 $$x-es$$ 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.

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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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08 Apr 2011, 07:28
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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04 Jul 2012, 03:55
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: 60605
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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04 Jul 2012, 03:59
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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21 Aug 2012, 16:51
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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19 May 2013, 00:51
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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Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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19 May 2013, 04:30
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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Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1  [#permalink]

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17 Oct 2013, 04:46
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
Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1   [#permalink] 17 Oct 2013, 04:46

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