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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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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, 21:50
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.

Hi Karishma,

For option A, i took, X=10

then for x^2-2x+A>0, A can be positive , negative or eve zero.

100-20+A>0

how could you decide if A is positive??
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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, 21:57
bsahil 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.

Hi Karishma,

For option A, i took, X=10

then for x^2-2x+A>0, A can be positive , negative or eve zero.

100-20+A>0

how could you decide if A is positive??

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.
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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, 23:23
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...??
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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
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, 21:24
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 ??
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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 Aug 2015, 06:50
samichange wrote:

Hi Bunnel

How do we solve (2) using the discriminant method?

thanks

I have a different take as to why statement 2 is not sufficient

y = -2x^1+1 will give you y >0 (=0.5) for x = 0.5. Thus Bunuel 's explanation that A$$\geq$$ 0 for y>0 for all x is not the complete explanation, IMHO.

Additionally, for A$$\geq$$0, y >0 for all x>0. Thus you get 2 different answers for "is A>0" based on these 2 cases and hence statement 2 is not sufficient.

samichange, using the discriminant method is not the best method for this statement.
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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 Aug 2015, 07:03
Engr2012 wrote:
samichange wrote:

Hi Bunnel

How do we solve (2) using the discriminant method?

thanks

I have a different take as to why statement 2 is not sufficient

y = -2x^1+1 will give you y >0 (=0.5) for x = 0.5. Thus Bunuel 's explanation that A$$\geq$$ 0 for y>0 for all x is not the complete explanation, IMHO.

Additionally, for A$$\geq$$0, y >0 for all x>0. Thus you get 2 different answers for "is A>0" based on these 2 cases and hence statement 2 is not sufficient.

samichange, using the discriminant method is not the best method for this statement.

Hi

Thanks for your analysis but what I understood is the following-

Consider x =0 and then whatever be the value of a ( + , - or 0 ), it does not matter because y = 0 + 1 ---> y > 0 and hence B is insufficient as a can assume any value.
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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 Aug 2015, 07:07
1
samichange wrote:
Engr2012 wrote:
samichange wrote:

Hi Bunnel

How do we solve (2) using the discriminant method?

thanks

I have a different take as to why statement 2 is not sufficient

y = -2x^1+1 will give you y >0 (=0.5) for x = 0.5. Thus Bunuel 's explanation that A$$\geq$$ 0 for y>0 for all x is not the complete explanation, IMHO.

Additionally, for A$$\geq$$0, y >0 for all x>0. Thus you get 2 different answers for "is A>0" based on these 2 cases and hence statement 2 is not sufficient.

samichange, using the discriminant method is not the best method for this statement.

Hi

Thanks for your analysis but what I understood is the following-

Consider x =0 and then whatever be the value of a ( + , - or 0 ), it does not matter because y = 0 + 1 ---> y > 0 and hence B is insufficient as a can assume any value.

Yes, you are correct about your explanation of why B is not sufficient and why discriminant method is not the wisest choice for this statement. Statement 1 was straightforward with the discriminant method as it would have remained a quadratic equation no matter what the value of A was but statement 2 will give an equation of a line if 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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09 Dec 2015, 07:04
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Is A positive?

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

There is one variable (A) and 2 equations are given by the conditions, so there is high chance (D) will become the answer.
For condition 1, from y=ax^2+bx+c, if D(discriminant)=b^2-4ac<0, y>0 works for all a>0.
From y=x^2-2x+A, D=(-2)^2-4*1*A<0, 4<4A, 1<A, which answers the question 'yes', makes the condition sufficient.
condition 2 answers the question 'yes' for A=1, but 'no' for A=0, making the condition insufficient.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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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 Jun 2018, 07:11
noboru wrote:
Is A positive?

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

1)$$x^2-2x+A$$ is positive for all x
This becomes x^2-2x+A>0
Most if the above solutions talk of Quadratic equation but equation are of form ax^2-2x+A=0 and it is easy to get confused in discriminant etc..
So a simplified way of looking at this..
$$x^2-2x+A>0$$ for all values of x....
So least value of (x^2-2x)+A>0

So what is least value ...
It is at x=1, x^2-2x=1^2-1*2=-1
So -1+A>0...A>1
Hence sufficient..
2) Ax^2+1>0
Ax^2>-1...
x^2 is always positive for all values so A can take both negative and positive values..
Insufficient

A
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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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26 Jun 2018, 17:39
(1)
The expression $$x^2 - 2x + A$$ is positive for *all* x. This means we can select any x we want and the expression must always be positive. So let's select an easy value for x, x=0. Plugging x=0 into the expression, we get
$$0^2 - 2(0) + A$$
So A is positive. If A were 0 or negative, then the given expression would not be positive for all x.
Sufficient

(2)
Same logic as for (1). Plugging in x=0, we get
$$A(0)^2 + 1$$
So A can take any value and the expression would be positive.
Not sufficient

Re: Is A positive? (1) x^2 - 2x + A is positive for all x (2) Ax^2 + 1 &nbs [#permalink] 26 Jun 2018, 17:39

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