It is currently 20 Jan 2018, 23:06

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:

### Hide Tags

Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 4695

Kudos [?]: 3323 [0], given: 0

GPA: 3.82
Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

### Show Tags

09 Dec 2015, 06:04
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.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Find a 10% off coupon code for GMAT Club members.
“Receive 5 Math Questions & Solutions Daily”
Unlimited Access to over 120 free video lessons - try it yourself
See our Youtube demo

Kudos [?]: 3323 [0], given: 0

Intern
Joined: 24 Dec 2015
Posts: 1

Kudos [?]: [0], given: 0

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

### Show Tags

09 Jan 2016, 22:59
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

Kudos [?]: [0], given: 0

Intern
Joined: 27 Oct 2015
Posts: 20

Kudos [?]: 2 [0], given: 8

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

### Show Tags

11 Dec 2016, 06:43
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

Kudos [?]: 2 [0], given: 8

SVP
Joined: 12 Dec 2016
Posts: 1860

Kudos [?]: 183 [0], given: 2406

Location: United States
GMAT 1: 700 Q49 V33
GPA: 3.64
Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

### Show Tags

07 Jul 2017, 12:18
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

Kudos [?]: 183 [0], given: 2406

Intern
Joined: 16 Apr 2017
Posts: 16

Kudos [?]: 0 [0], given: 112

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

### Show Tags

07 Jul 2017, 23:17
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.

Kudos [?]: 0 [0], given: 112

SVP
Joined: 12 Dec 2016
Posts: 1860

Kudos [?]: 183 [0], given: 2406

Location: United States
GMAT 1: 700 Q49 V33
GPA: 3.64
Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is [#permalink]

### Show Tags

08 Jul 2017, 19:43
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.

Kudos [?]: 183 [0], given: 2406

Intern
Joined: 06 Apr 2017
Posts: 27

Kudos [?]: 14 [0], given: 38

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

### Show Tags

01 Aug 2017, 12:17
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!

Kudos [?]: 14 [0], given: 38

Re: Is A positive? x^2-2x+A is positive for all x Ax^2+1 is   [#permalink] 01 Aug 2017, 12:17

Go to page   Previous    1   2   3   4   [ 67 posts ]

Display posts from previous: Sort by

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

 new topic post reply Question banks Downloads My Bookmarks Reviews Important topics

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.