Data Sufficiency

Is A positive?

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

my answer is E and here is how i solved it.
statement 1. if x=-3 so 9+6+a>0 so A could be negative or positive so statement is insufficient
statement2: if x=1 and A>0 so ok, A >0 but if x=1 and A=-1/2 so -1/2+1=1/2>0 so A could be negative or positive thus statement 2 is not sufficient
statement 1+2: x=1 so 1-2+A>0 so A>0
if x=-1 so 1+2+a>0 so a could be positive or negative
so the answer is E
where am i wrong?

I overlooked the use of "0" as well.

Cheers,

diebeatsthegmat wrote:Is A positive?

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

Hi!

It's key to understand exactly what a statement is telling you.

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

To satisfy this statement, we have to have an A that makes the expression positive for EVERY POSSIBLE value of x.

Well, if x=0, then the expression simply equals A. So, in order for the statement to be positive when x=0, A must be positive. Therefore, to guarantee that the expression is ALWAYS positive, regardless of the value of x, we have to have a positive A: sufficient, eliminate B, C and E.

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

Again, we need a value of A that will guarantee that the expression is ALWAYS positive, regardless of the value of x.

Let's simplify the inequality:

A(x^2) + 1 > 0
A(x^2) > -1

If x^2 = 0, then the expression will be true.

x^2 can never be negative, so putting 0 aside, it's safe to divide both sides by x^2:

A > -1/x^2

Since A must be bigger than a negative number, A could be negative, 0 or positive: insufficient, eliminate (D).

You could also just pick numbers:

if A=0, then the statement will always hold true (since 0*anything is greater than -1).
if A=1, then the statement will always hold true (since 1*(non negative) is greater than or equal to 0 which is greater than -1).

Since A=1 gives us a yes answer and A=0 gives us a no answer to "is A positive?", (2) is insufficient.

(1) is sufficient, (2) isn't: choose (A)!

Zero makes sense.

Stuart Kovinsky wrote:

diebeatsthegmat wrote:Is A positive?

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

Hi!

It's key to understand exactly what a statement is telling you.

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

To satisfy this statement, we have to have an A that makes the expression positive for EVERY POSSIBLE value of x.

Well, if x=0, then the expression simply equals A. So, in order for the statement to be positive when x=0, A must be positive. Therefore, to guarantee that the expression is ALWAYS positive, regardless of the value of x, we have to have a positive A: sufficient, eliminate B, C and E.

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

Again, we need a value of A that will guarantee that the expression is ALWAYS positive, regardless of the value of x.

Let's simplify the inequality:

A(x^2) + 1 > 0
A(x^2) > -1

If x^2 = 0, then the expression will be true.

x^2 can never be negative, so putting 0 aside, it's safe to divide both sides by x^2:

A > -1/x^2

Since A must be bigger than a negative number, A could be negative, 0 or positive: insufficient, eliminate (D).

You could also just pick numbers:

if A=0, then the statement will always hold true (since 0*anything is greater than -1).
if A=1, then the statement will always hold true (since 1*(non negative) is greater than or equal to 0 which is greater than -1).

Since A=1 gives us a yes answer and A=0 gives us a no answer to "is A positive?", (2) is insufficient.

(1) is sufficient, (2) isn't: choose (A)!

thank you!