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

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

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New post 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.
The answer therefore becomes (A).

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

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New post 09 Jan 2016, 23:59
From I
if x=3 and a=1/2 it is +ve
if x=3 and a=-1/2 still is +ve
so insufficient

From II
if x=1 and a=1/2 it is +ve
If x=1 and a=-1/2 it is +ve
so both insufficient for same values so E is correct.
Please correct me if i am wrong

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

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New post 11 Dec 2016, 07:43
Hi Buenel,
The point here is that x2−2x+A>0x2−2x+A>0 for all x−esx−es.

Let's do this in another way:

We have (x2−2x)+A>0(x2−2x)+A>0 for all x−esx−es. The sum of 2 quantities (x2−2xx2−2x and AA) is positive for all x−esx−es. So for the least value of x2−2xx2−2x, AA must make the whole expression positive.

So what is the least value of x2−2xx2−2x? The least value of quadratic expression ax2+bx+cax2+bx+c is when x=−b2ax=−b2a, so in our case the least value of x2−2xx2−2x is when x=−−22=1x=−−22=1 --> x2−2x=−1x2−2x=−1 --> −1+A>0−1+A>0 --> A>1A>1.

OR:

x2−2x+A>0x2−2x+A>0 --> x2−2x+1+A−1>0x2−2x+1+A−1>0 --> (x−1)2+A−1>0(x−1)2+A−1>0 --> least value of (x−1)2(x−1)2 is zero thus A−1A−1 must be positive (0+A−1>00+A−1>0)--> A−1>0A−1>0 --> A>1A>1.

My Question is why should the least value of (x-1)^2 be 1?

Not understanding the same

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

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New post 07 Jul 2017, 13:18
In st 2 is still true if A = 0 => B and D and C are all out
In st 1, (x-1)^2 +A >= A with all x
Therefore "x^2-2x+A is positive for all x" can be true only if A >= 1 => A is positive

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

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New post 08 Jul 2017, 00:17
Bunuel wrote:
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.



Dear Bunuel,

What if x is 3 and A is -2? The expression in option 1 is still positive.

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

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New post 08 Jul 2017, 20:43
SataC wrote:
Bunuel wrote:
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.



Dear Bunuel,

What if x is 3 and A is -2? The expression in option 1 is still positive.

You miss an important point in BB's solution; that is why you misread everything. The solution is to find the condition of A so that the option 1 is ALWAYS positive with ALL X.

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

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New post 01 Aug 2017, 13:17
Bunuel,

I did the same thing as Diana for statement 2)

Using an equation of the form:

\(Ax^2+Bx+C\)

If \(B^2-4AC < 0\),

then there are no real roots. Statement 2 is essentially telling us that there are no real roots, so I set this up with:

\(A=a\)
\(B=0\)
\(C=1\)

\(0^2-4a < 0\)
\(a>0\)

Now, I understand the pure logic of your explanation for statement 2, but I'm wondering why exactly the discriminant is providing the wrong answer when \(B=0\). I want to avoid similar assumption considering roots and the discriminant in the future. Thanks!

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Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is   [#permalink] 01 Aug 2017, 13:17

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