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Re: Is A positive? [#permalink]
15 Jul 2010, 13:40

9

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Expert's post

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.

Re: Is A positive? [#permalink]
16 Jul 2010, 07:01

8

This post received KUDOS

Expert's post

1

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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>0for allx-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.

Re: Is A positive? [#permalink]
16 Jul 2010, 08:49

Expert's post

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 anyx 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.

Re: Is A positive? [#permalink]
16 Jul 2010, 08:56

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 anyx 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.

Thanks for the reply Bunuel,

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.

Re: Is A positive? [#permalink]
16 Jul 2010, 09:16

Expert's post

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 anyx 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.

Thanks for the reply Bunuel,

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>0FOR 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. _________________

Re: DS: number properties [#permalink]
10 Apr 2012, 20:03

1) X^2-2X+A is positive for all X I think A could not be the answer, for example, if A = 0, and X = 4, then also the expression is positive, but A = 0 is neither positive nor negative Again, if A = 1, and and X = 4, then also the expression is positive 2) AX^2 + 1 is positive for all X Same logic as above, if A is 0, then the expression is positive, and the expression is also postive for any value of X where A > 0

In a nutshell, I too think the answer is E. _________________

Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

Re: DS: number properties [#permalink]
11 Apr 2012, 09:42

1

This post received KUDOS

Expert's post

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.

Re: Is A positive? [#permalink]
04 Jul 2012, 02:59

Expert's post

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.

Answer: A.

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.

Is A positive? 1. x^2 -2x +A is positive for all x. 2. A*x^2 +1 is positive for all x.

I got E and my way of solving is as below: St1. x^2 - 2x +A > 0 Let x=0, so A>0. Let x=-1, so A>-3. In this case A can be negative or positive. Insufficient.

St2. A*x^2 +1 > 0 Let x=-1, so A>-1. Again A can be positive or negative. Insufficient.

St1+St2: Let x=-1, so A > -1. Again A can be positive or negative. Insufficient.

So it's E. However the OA is not E. Please advise.

Is A positive? 1. x^2 -2x +A is positive for all x. 2. A*x^2 +1 is positive for all x.

I got E and my way of solving is as below: St1. x^2 - 2x +A > 0 Let x=0, so A>0. Let x=-1, so A>-3. In this case A can be negative or positive. Insufficient.

St2. A*x^2 +1 > 0 Let x=-1, so A>-1. Again A can be positive or negative. Insufficient.

St1+St2: Let x=-1, so A > -1. Again A can be positive or negative. Insufficient.

So it's E. However the OA is not E. Please advise.

Merging similar topics. Please refer to the solutions above. _________________

Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]
21 Aug 2012, 15:51

1

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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.... _________________