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

 It is currently 27 Jun 2019, 01:29 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  M03-18

Author Message
TAGS:

Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

2
5 00:00

Difficulty:   15% (low)

Question Stats: 71% (00:54) correct 29% (00:52) wrong based on 233 sessions

HideShow timer Statistics How many distinct roots does the equation $$x^4 - 2x^2 + 1=0$$ have?

A. 0
B. 1
C. 2
D. 3
E. 4

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

1
Official Solution:

How many distinct roots does the equation $$x^4 - 2x^2 + 1=0$$ have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$x^4 - 2x^2 +1=0$$;

$$(x^2-1)^2=0$$;

$$x^2-1=0$$;

$$(x-1)(x+1)=0$$. Either $$x=1$$ or $$x=-1$$. So, the given equation has two distinct roots.

_________________
Intern  Joined: 06 Apr 2014
Posts: 2

Show Tags

Value of x cant be negative as it equals to the even (4th) root of some expression.

So ,according to me x=1,therefore only one root.

Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

gmattaker10 wrote:
Value of x cant be negative as it equals to the even (4th) root of some expression.

So ,according to me x=1,therefore only one root.

That's not correct. Plug -1 into the equation, does it hold true?
_________________
Manager  S
Joined: 03 Aug 2015
Posts: 52
Concentration: Strategy, Technology
Schools: ISB '18, SPJ GMBA '17
GMAT 1: 680 Q48 V35 Show Tags

1
Bunuel wrote:
Official Solution:

How many distinct roots does the equation $$x^4 - 2x^2 + 1=0$$ have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$x^4 - 2x^2 +1=0$$;

$$(x^2-1)^2=0$$;

$$x^2-1=0$$;

$$(x-1)(x+1)=0$$. Either $$x=1$$ or $$x=-1$$. So, the given equation has two distinct roots.

Bunel,

Could you pls explain the steps to get to the highlighted step?

Thanks,
A
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

ArunpriyanJ wrote:
Bunuel wrote:
Official Solution:

How many distinct roots does the equation $$x^4 - 2x^2 + 1=0$$ have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$x^4 - 2x^2 +1=0$$;

$$(x^2-1)^2=0$$;

$$x^2-1=0$$;

$$(x-1)(x+1)=0$$. Either $$x=1$$ or $$x=-1$$. So, the given equation has two distinct roots.

Bunel,

Could you pls explain the steps to get to the highlighted step?

Thanks,
A

It's a simple algebraic property: $$a^2-2ab+b^2=(a-b)^2$$
_________________
Intern  Joined: 18 May 2016
Posts: 6

Show Tags

Bunuel wrote:
Official Solution:

How many distinct roots does the equation $$x^4 - 2x^2 + 1=0$$ have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$x^4 - 2x^2 +1=0$$;

$$(x^2-1)^2=0$$;

[b]$$x^2-1=0$$;

$$(x-1)(x+1)=0$$. Either $$x=1$$ or $$x=-1$$. So, the given equation has two distinct roots.

can you please explain how you eliminated the square in the highlighted step. thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

bjklue wrote:
Bunuel wrote:
Official Solution:

How many distinct roots does the equation $$x^4 - 2x^2 + 1=0$$ have?

A. 0
B. 1
C. 2
D. 3
E. 4

$$x^4 - 2x^2 +1=0$$;

$$(x^2-1)^2=0$$;

[b]$$x^2-1=0$$;

$$(x-1)(x+1)=0$$. Either $$x=1$$ or $$x=-1$$. So, the given equation has two distinct roots.

can you please explain how you eliminated the square in the highlighted step. thanks

This is basics: number^2 = 0 --> number = 0.
_________________
Manager  S
Joined: 23 Jan 2016
Posts: 180
Location: India
GPA: 3.2

Show Tags

Bunuel, though this may be a simple question, I always struggle to spot and close an open quadratic to a closed one, as has been done in this question. Is there any simple way to do so?
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

OreoShake wrote:
Bunuel, though this may be a simple question, I always struggle to spot and close an open quadratic to a closed one, as has been done in this question. Is there any simple way to do so?

I guess the only way is through practice...
_________________
Intern  B
Joined: 04 Apr 2017
Posts: 17

Show Tags

In the given equation, substitute y = x^2.
The equation will reduce to a quadratic equation and the roots of y will 1.

Therefore, x^2 = 1 which means, x can be +1 or -1.
Retired Moderator P
Joined: 19 Mar 2014
Posts: 929
Location: India
Concentration: Finance, Entrepreneurship
GPA: 3.5

Show Tags

Equation: x^4 - 2x^2 + 1=0

(x^2 - 1)^2 = 0

(x^2 - 1) = 0

x^2 = 1

Hence, x = +/- 1

So, this Equation will have two distinct roots

_________________
"Nothing in this world can take the place of persistence. Talent will not: nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not: the world is full of educated derelicts. Persistence and determination alone are omnipotent."

Best AWA Template: https://gmatclub.com/forum/how-to-get-6-0-awa-my-guide-64327.html#p470475
Manager  G
Joined: 21 Mar 2017
Posts: 136
Location: India
GMAT 1: 560 Q48 V20 WE: Other (Computer Software)

Show Tags

Bunuel,

I rejected -1 because (x2−1)2=0(x2−1)2=0.

Could you please provide few links to get a firm grip on such questions.

I have seen one solution before in which one value was eliminated because it was raised to the power of 4.
_________________
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
When nothing seem to help, I would go and look at a Stonecutter hammering away at his rock perhaps a hundred time without as much as a crack showing in it.
Yet at the hundred and first blow it would split in two.
And I knew it was not that blow that did it, But all that had gone Before
.
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

Prashant10692 wrote:
Bunuel,

I rejected -1 because (x2−1)2=0(x2−1)2=0.

Could you please provide few links to get a firm grip on such questions.

I have seen one solution before in which one value was eliminated because it was raised to the power of 4.

7. Algebra

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
_________________
Intern  B
Joined: 07 Apr 2018
Posts: 6

Show Tags

I factored x^2:
x^2 (x^2 - 1) = 0
Then I get three solutions:
x^2=0, i.e. x=0
x^2=1, i.e. x=1 and x=-1

Is it in general possible to factor in that way and, if so, where's the flaw?
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

1
gmatprep001 wrote:
I factored x^2:
x^2 (x^2 - 1) = 0
Then I get three solutions:
x^2=0, i.e. x=0
x^2=1, i.e. x=1 and x=-1

Is it in general possible to factor in that way and, if so, where's the flaw?

You would be correct if we had x^4 - x^2 = 0 but we have $$x^4 - 2x^2 + 1=0$$.
_________________
Senior Manager  P
Joined: 16 Nov 2016
Posts: 274

Show Tags

Hi Bunuel

I did this:

$$x^2(x^2 - 2) = -1$$

$$x^2 = 1$$

therefore x has two roots 1 and -1

is this approach correct?
_________________
If you find my post useful, please give me a kudos.

Thank you.
Regards,
ENEM

If you wish to spend wisely on your gmat prep material, check my post titled: How to Spend Money On GMAT Material Wisely, link: https://gmatclub.com/forum/how-to-buy-gmat-material-wisely-tag-free-gmat-resources-236174.html

Simple and handy template for CR: https://gmatclub.com/forum/simple-and-handy-template-for-cr-242255.html

simple template for more vs greater and fewer vs less: https://gmatclub.com/forum/simple-template-for-more-vs-greater-and-fewer-vs-less-242216.html
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

1
ENEM wrote:
Hi Bunuel

I did this:

$$x^2(x^2 - 2) = -1$$

$$x^2 = 1$$

therefore x has two roots 1 and -1

is this approach correct?

The answer is correct. Not clear though how you identified that x^2 = 1 from x^2(x^2 - 2) = -1 or how would you get the answer if it were x^2(x^2 - 2) = 10.
_________________
Senior Manager  P
Joined: 16 Nov 2016
Posts: 274

Show Tags

Bunuel wrote:
ENEM wrote:
Hi Bunuel

I did this:

$$x^2(x^2 - 2) = -1$$

$$x^2 = 1$$

therefore x has two roots 1 and -1

is this approach correct?

The answer is correct. Not clear though how you identified that x^2 = 1 from x^2(x^2 - 2) = -1 or how would you get the answer if it were x^2(x^2 - 2) = 10.

Bunuel

$$x^2(x^2 − 2)= −1$$

$$x^2 = −1$$ and$$(x^2−2) = −1$$

this $$x^2 = −1$$ is not possible as square of a number cannot be negative.

thus we are left with $$x^2−2 = −1$$

adding 2 on both sides we get

$$x^2=1$$

therefore x has two roots 1 and -1

did I goof up??
_________________
If you find my post useful, please give me a kudos.

Thank you.
Regards,
ENEM

If you wish to spend wisely on your gmat prep material, check my post titled: How to Spend Money On GMAT Material Wisely, link: https://gmatclub.com/forum/how-to-buy-gmat-material-wisely-tag-free-gmat-resources-236174.html

Simple and handy template for CR: https://gmatclub.com/forum/simple-and-handy-template-for-cr-242255.html

simple template for more vs greater and fewer vs less: https://gmatclub.com/forum/simple-template-for-more-vs-greater-and-fewer-vs-less-242216.html
Math Expert V
Joined: 02 Sep 2009
Posts: 55804

Show Tags

1
ENEM wrote:
Bunuel wrote:
ENEM wrote:
Hi Bunuel

I did this:

$$x^2(x^2 - 2) = -1$$

$$x^2 = 1$$

therefore x has two roots 1 and -1

is this approach correct?

The answer is correct. Not clear though how you identified that x^2 = 1 from x^2(x^2 - 2) = -1 or how would you get the answer if it were x^2(x^2 - 2) = 10.

Bunuel

x^2(x^2 − 2)= −1

x^2 = −1 and (x^2−2) = −1

this x^2 = −1 is not possible as square of a number cannot be negative.

thus we are left with x^2−2 = −1

adding 2 on both sides we get

x^2=1

therefore x has two roots 1 and -1

did I goof up??

Yes, that's not correct you got the correct answer by fluke. Check my response here: https://gmatclub.com/forum/m03-183603.html#p2050204
_________________ Re: M03-18   [#permalink] 15 Sep 2018, 05:28

Go to page    1   2    Next  [ 24 posts ]

Display posts from previous: Sort by

M03-18

Moderators: chetan2u, Bunuel  