It is currently 22 Jun 2017, 10:13

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

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

M12 Q17

Author Message
Manager
Status: A continuous journey of self-improvement is essential for every person -Socrates
Joined: 02 Jan 2011
Posts: 71

Show Tags

02 Mar 2011, 20:41
IanStewart wrote:
durgesh79 wrote:
How many roots does this equation have?

$$sqrt(x^2+1) + sqrt(x^2+2) = 2$$

0
1
2
3
4

I don't think we need to do any algebra here: $$x^2 \geq 0$$, so $$\sqrt{x^2+1} + \sqrt{x^2+2} \geq 1 + \sqrt{2}$$, which is larger than 2. Hence no (real) solutions for x.

Please explain that how we have assumed $$x^2 \geq 0$$ at the start of the problem and what is the theory behind this assumption. I have also solved with the long method as mentioned by Bunnel.

SVP
Joined: 16 Nov 2010
Posts: 1665
Location: United States (IN)
Concentration: Strategy, Technology

Show Tags

02 Mar 2011, 21:12
x^2 >= 0 as square of a real number is never negative.
_________________

Formula of Life -> Achievement/Potential = k * Happiness (where k is a constant)

GMAT Club Premium Membership - big benefits and savings

Senior Manager
Joined: 01 Nov 2010
Posts: 288
Location: India
Concentration: Technology, Marketing
GMAT Date: 08-27-2012
GPA: 3.8
WE: Marketing (Manufacturing)

Show Tags

03 Mar 2011, 22:52
1
KUDOS
u don't need to calculate anything.

clearly, x^2 >= 0.
sqrt 2 = 1.414
so, SQRT(x^2+1) + SQRT(x^2+2) = 2,

it doesn't have any solution beacuse left hand side is always greater than 2.
the min value of left hand side is 2.414. which is greater than 2.
_________________

kudos me if you like my post.

Attitude determine everything.
all the best and God bless you.

Manager
Joined: 02 Feb 2012
Posts: 204
Location: United States
GMAT 1: 770 Q49 V47
GPA: 3.08

Show Tags

05 Mar 2012, 09:33
Got the question wrong, just went too quickly and didn't really think it through. Reading the explanations makes complete sense though.
Intern
Joined: 15 Apr 2011
Posts: 49

Show Tags

05 Mar 2012, 23:58
Took me a while but in for (A)...
Although... I feel that this might be out of scope question for GMAT... Right?
Manager
Status: I will not stop until i realise my goal which is my dream too
Joined: 25 Feb 2010
Posts: 229
Schools: Johnson '15

Show Tags

26 Apr 2012, 05:06
1
KUDOS
Bunuel wrote:
TheSituation wrote:
I got A but solved thru a different (less clever) method. I think I may have answered correct in spite of myself though, can someone tell me if my solution is mathmatically sound or if I arrived at the correct solution by luck.

Begin by squaring both sides

(x^2+1) + (x^2 +2) = 4
drop brackets and solve
2x^2=1
x^2=1/2
x = sq root of 1/2
therefore no real roots, therefore A

feedback?

This would be the longer way, plus you'll need to square twice not once, as you made a mistake while squaring first time.

$$(a+b)^2=a^2+2ab+b^2$$, so when you square both sides you'll get: $$x^2+1+2\sqrt{(x^2+1)(x^2 +2)}+x^2+2= 4$$ --> $$2\sqrt{(x^2+1)(x^2 +2)}=1-2x^2$$. At this point you should square again --> $$4x^4+12x^2+8=1-4x^2+4x^4$$ --> $$16x^2=-7$$, no real $$x$$ satisfies this equation.

There is one more problem with your solution:

You've got (though incorrectly): $$x^2=\frac{1}{2}$$ and then you concluded that this equation has no real roots, which is not right. This quadratic equation has TWO real roots: $$x=\frac{1}{\sqrt{2}}$$ and $$x=-{\frac{1}{\sqrt{2}}}$$. Real roots doesn't mean that the roots must be integers, real roots means that roots must not be complex numbers, which I think we shouldn't even mention as GMAT deals ONLY with real numbers. For example $$x^2=-1$$ has no real roots and for GMAT it means that this equation has no roots, no need to consider complex roots and imaginary numbers.

Hope it's clear.

this is the kind of questions i had got in real GMAT and i was stumped...took 5 mins, yet no correct answer.. .now i am slowly understanding how to answer these kinds
_________________

Regards,
Harsha

Note: Give me kudos if my approach is right , else help me understand where i am missing.. I want to bell the GMAT Cat

Satyameva Jayate - Truth alone triumphs

Intern
Joined: 10 Aug 2012
Posts: 19
Location: India
Concentration: General Management, Technology
GPA: 3.96

Show Tags

06 Mar 2013, 22:48
1
KUDOS
Given equation is

$$\sqrt{(x^2 +1)} + \sqrt{(x^2 +2)} = 2$$

As we know that $$x^2$$ is greater than zero for all values of $$x$$

So the least possible value of LHS of equation will be when$$x = 0$$

At $$x = 0$$, LHS = $$\sqrt{(0 +1)} + \sqrt{(0 +2)} >$$ RHS $$(2)$$

So equation does not have any real root.....
Verbal Forum Moderator
Joined: 16 Jun 2012
Posts: 1130
Location: United States

Show Tags

08 Mar 2013, 00:02
Tricky question. After 2 minutes I had to pick answer randomly and got this question wrong... I saw because x^2 >= 0, so the left side is always > 2. Very nice question.
_________________

Please +1 KUDO if my post helps. Thank you.

"Designing cars consumes you; it has a hold on your spirit which is incredibly powerful. It's not something you can do part time, you have do it with all your heart and soul or you're going to get it wrong."

Chris Bangle - Former BMW Chief of Design.

Manager
Joined: 20 Oct 2013
Posts: 76
Location: United States
Concentration: General Management, Real Estate

Show Tags

27 Apr 2014, 00:58
x^2+1>=1 and x^2+2>=2 so the left side >= sqrt(1)+sqrt(2) >2sqrt(1) = 2

Hence the equation has no root.
Senior Manager
Status: You have to have the darkness for the dawn to come
Joined: 09 Nov 2012
Posts: 279
Location: India
saurav: suman
Concentration: Operations, Technology
GPA: 4
WE: Engineering (Energy and Utilities)

Show Tags

05 May 2014, 13:01
IanStewart wrote:
durgesh79 wrote:
How many roots does this equation have?

$$sqrt(x^2+1) + sqrt(x^2+2) = 2$$

0
1
2
3
4

I don't think we need to do any algebra here: $$x^2 \geq 0$$, so $$\sqrt{x^2+1} + \sqrt{x^2+2} \geq 1 + \sqrt{2}$$, which is larger than 2. Hence no (real) solutions for x.

The solution explained by you is just awesome.
Though I solved the problem but I use lot of algebra and took much time to solve the problem.
Nice explanation
_________________

You have to have the darkness for the dawn to come.

Give Kudos if you like my post

Re: M12 Q17   [#permalink] 05 May 2014, 13:01

Go to page   Previous    1   2   [ 30 posts ]

Similar topics Replies Last post
Similar
Topics:
6 M17 Q 17 2 07 Jul 2014, 17:14
m10, q17 0 06 Jul 2010, 19:51
1 M10 Q17 11 16 Jun 2012, 10:47
Verbal Test 5, Q17 0 28 Dec 2009, 10:08
19 m01 Q17 18 12 Jul 2013, 07:04
Display posts from previous: Sort by

M12 Q17

Moderator: Bunuel

 Powered by phpBB © phpBB Group and phpBB SEO 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®.