GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 12 Dec 2019, 01:41

### 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

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

# a and b are two numbers such that 2a^4 -4ab + b^2 + 2 = 0...

Author Message
TAGS:

### Hide Tags

Manager
Joined: 21 Feb 2019
Posts: 125
Location: Italy
a and b are two numbers such that 2a^4 -4ab + b^2 + 2 = 0...  [#permalink]

### Show Tags

18 Apr 2019, 05:51
3
00:00

Difficulty:

85% (hard)

Question Stats:

39% (02:25) correct 61% (02:08) wrong based on 28 sessions

### HideShow timer Statistics

$$a$$ and $$b$$ are two numbers such that:

$$2a^4 -4ab + b^2 + 2 = 0$$

How many distinct values can $$a$$ assume?

A. 1
B. 2
C. 3
D. 4
E. None
Math Expert
Joined: 02 Aug 2009
Posts: 8305
Re: a and b are two numbers such that 2a^4 -4ab + b^2 + 2 = 0...  [#permalink]

### Show Tags

20 Apr 2019, 08:03
1
lucajava wrote:
$$a$$ and $$b$$ are two numbers such that:

$$2a^4 -4ab + b^2 + 2 = 0$$

How many distinct values can $$a$$ assume?

A. 1
B. 2
C. 3
D. 4
E. None

$$2a^4 -4ab + b^2 + 2 = 0$$.....$$2a^4-4a^2+4a^2 -4ab + b^2 + 2 = 0.....2(a^4-2a^2+1)+(4a^2-4ab+b^2)=0....2(a^2-1)^2+(2a-b)^2=0$$...
Since sum of two squares will be 0 when each term is 0..
2(a^2-1)^2=0, that is a=-1 or a=1.
(2a-b)^2=0, that is a=b/2..
When a=-1.. $$2a^4 -4ab + b^2 + 2 = 0.....2*(-1)^4-4*(-1)b+b^2+2=0...2+4b+b^2+2=0....(b+2)^2=0$$..(a,b)=(-1,-2)..
When a=1.. $$2a^4 -4ab + b^2 + 2 = 0.....2*(1)^4-4*(1)b+b^2+2=0...2-4b+b^2+2=0....(b-2)^2=0$$..(a,b)=(1,2)..

So a has two distinct values -1 and 1..

B
_________________
Manager
Joined: 21 Feb 2019
Posts: 125
Location: Italy
Re: a and b are two numbers such that 2a^4 -4ab + b^2 + 2 = 0...  [#permalink]

### Show Tags

21 Apr 2019, 13:18
Another way is considering $$a$$ a fixed parameter. Hence, you can consider:

$$b^2 -4ab +2a^4 + 2 = 0$$

a second-degree equation. Let's compute $$b_{12} = 2a ± \sqrt{4a^2 -2a^4 -2}$$.

Since GMAT math works on real numbers only, $$4a^2 -2a^4 - 2 ≥ 0$$. Hence, developing this, we get:

$$-a^4 +2a^2 -1 ≥ 0$$

$$a^4 -2a^2 +1 ≤ 0$$

Let be $$a^2 = t$$

$$t^2 -2t + 1 ≤ 0$$

$$t_{12} = 1 ± \sqrt{1 -1} = 1$$

In conclusion, we derive $$a^2 = t = 1$$. Hence, $$a = ± 1$$. Two solutions, pick B.
Re: a and b are two numbers such that 2a^4 -4ab + b^2 + 2 = 0...   [#permalink] 21 Apr 2019, 13:18
Display posts from previous: Sort by