Last visit was: 09 Sep 2024, 07:56 It is currently 09 Sep 2024, 07:56
Decision Tracker
My Rewards
Unanswered
Active Topics
New posts
New comers' posts
Close
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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

If x+sqrt(x^2-4x+4)=2, then:

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
P
fskilnik
Tutor
Joined: 12 Oct 2010
Status:GMATH founder
Posts: 891
Own Kudos [?]: 1444 [20]
Given Kudos: 56
Send PM
If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   Updated on: 27 Mar 2019, 12:52
20
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

95% (hard)

Question Stats:

27% (01:47) correct 73% (01:53) wrong based on 211 sessions
Hide Show DIFFICULTY AND TIMER STATISTICS
GMATH practice exercise (Quant Class 12)



Show Spoiler
The alternative choices are tricky... the official answer is NOT wrong, and you can find it in 3 SECONDS, using LOGIC only!
ShowHide Answer
Official Answer
A

Originally posted by fskilnik on 27 Mar 2019, 12:44.
Last edited by fskilnik on 27 Mar 2019, 12:52, edited 1 time in total.
Most Helpful Reply
User avatar
P
fskilnik
Tutor
Joined: 12 Oct 2010
Status:GMATH founder
Posts: 891
Own Kudos [?]: 1444 [7]
Given Kudos: 56
Send PM
Re: If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   27 Mar 2019, 13:30
4
Kudos
3
Bookmarks
Expert Reply
fskilnik wrote:
GMATH practice exercise (Quant Class 12)



Hi, Ian! Thanks for joining, for your kind words and for all your important comments related to the "theme"!

Let me add the following observation:

In each question, the GMAT asks the test taker to find the BEST answer (among the 5 alternative choices offered), therefore if you were given the following alternative choices:

(A) x < 3
(B) x < 2
(C) ---
(D) ---
(E) ---

You would be supposed to consider (B) the "proper" right answer (it is more "restrictive"), although (as explained by Ian) alternative choice (A) is also true. (If x < 2 then x<3, of course!)

The fact is that I did NOT offer two right answers (to have to deal with the "best" one to be chosen, something I personally dislike), therefore the official answer is (A) without any mess!

Okay... I hope this comment is a good complement to Ian´s discussion!

Now let´s go to "the official solution":


\(?\,\,\,:\,\,\,\,x\,\,{\rm{must}}\,\,{\rm{be}}\,\,\, \ldots\)


\(x + \sqrt {{x^2} - 4x + 4} \,\,\, = \,\,\,2\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\sqrt {{{\left( {x - 2} \right)}^2}} = 2 - x\)

\(\,\,\,\, \Leftrightarrow \,\,\,\,\,\left| {x - 2} \right| = - \left( {x - 2} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x - 2 \le 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \le 2\)


\(x \le 2\,\,\,\, \Rightarrow \,\,\,\left( A \right)\,\,{\rm{is}}\,\,{\rm{true}}\,\,\,\,\left[ {{\rm{and}}\,\,\left( B \right),\left( C \right),\left( D \right),\left( E \right)\,\,{\rm{are}}\,\,{\rm{false}}} \right]\)


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
General Discussion
User avatar
P
fskilnik
Tutor
Joined: 12 Oct 2010
Status:GMATH founder
Posts: 891
Own Kudos [?]: 1444 [0]
Given Kudos: 56
Send PM
If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   27 Mar 2019, 12:56
Expert Reply
subhadeep13 wrote:
A
Sent from my iPhone using GMAT Club Forum

Excellent! I have edited my "hint" to mention the possibility of finding the right answer in less than a few seconds. Congrats!

Now I suggest you answer the question in the "traditional way"... :)

P.S.: if we take into account that GMAT asks the candidate to find the BEST answer, someone could argue that it is very "risky" to adopt the 3-seconds approach... think about that!
User avatar
L
IanStewart
GMAT Tutor
Joined: 24 Jun 2008
Posts: 4127
Own Kudos [?]: 9599 [3]
Given Kudos: 96
 Q51  V47
Send PM
Re: If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   27 Mar 2019, 13:10
3
Kudos
Expert Reply
Doing this in the mathematical way (the 3-second answer just requires you to notice that the interval answer A contains the interval in every other answer choice, so if one of the other answers were correct, A would naturally also need to be correct, and a GMAT problem cannot have two different correct answers) --

x^2 - 4x + 4 = (x-2)^2, so √(x^2 - 4x + 4) = √(x-2)^2

Many test takers, seeing something like √(a^2), will think that is equal to a. That's one of the more common errors test takers make on higher level GMAT algebra questions. It is true that √(a^2) = a when a is positive or zero. But it's not true if a is negative -- if you plug a = -3 into √(a^2), you'll see that it is equal to 3, so the sign has flipped from negative to positive. In general, all we can say is that √(a^2) = |a| is always true.

So √(x-2)^2 = |x-2|, and our equation here becomes:

x + |x - 2| = 2

We can solve this using cases:

- if the thing inside the absolute value, x-2, is positive or zero, the absolute value will do nothing, so we can erase it and solve. Then we get

x + x - 2 = 2
2x = 4
x = 2

So when x - 2 > 0, there is only one solution, x = 2.

- when x-2 is negative, though, the absolute value will flip it's sign, so it will become 2-x. So when x < 2, our equation becomes

x + 2 - x = 2
2 = 2

and the equation is always true for every value of x < 2.

Combining our solutions from the two cases, the equation will be true whenever x < 2. If that were an answer choice, it would be the right answer, but we don't find that among the choices. But if we know for certain that x < 2, then it's clearly true that x < 3, so that's the right answer here. We can't be certain any of the other answer choices are correct, because some potential values of x lie outside the intervals in every other choice.

Nice question!
avatar
B
anmolsd1995
Intern
Intern
Joined: 17 Apr 2017
Posts: 11
Own Kudos [?]: 3 [0]
Given Kudos: 26
Send PM
Re: If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   06 Jun 2020, 07:49
IanStewart wrote:
Doing this in the mathematical way (the 3-second answer just requires you to notice that the interval answer A contains the interval in every other answer choice, so if one of the other answers were correct, A would naturally also need to be correct, and a GMAT problem cannot have two different correct answers) --

x^2 - 4x + 4 = (x-2)^2, so √(x^2 - 4x + 4) = √(x-2)^2

Many test takers, seeing something like √(a^2), will think that is equal to a. That's one of the more common errors test takers make on higher level GMAT algebra questions. It is true that √(a^2) = a when a is positive or zero. But it's not true if a is negative -- if you plug a = -3 into √(a^2), you'll see that it is equal to 3, so the sign has flipped from negative to positive. In general, all we can say is that √(a^2) = |a| is always true.

So √(x-2)^2 = |x-2|, and our equation here becomes:

x + |x - 2| = 2

We can solve this using cases:

- if the thing inside the absolute value, x-2, is positive or zero, the absolute value will do nothing, so we can erase it and solve. Then we get

x + x - 2 = 2
2x = 4
x = 2

So when x - 2 > 0, there is only one solution, x = 2.

- when x-2 is negative, though, the absolute value will flip it's sign, so it will become 2-x. So when x < 2, our equation becomes

x + 2 - x = 2
2 = 2

and the equation is always true for every value of x < 2.

Combining our solutions from the two cases, the equation will be true whenever x < 2. If that were an answer choice, it would be the right answer, but we don't find that among the choices. But if we know for certain that x < 2, then it's clearly true that x < 3, so that's the right answer here. We can't be certain any of the other answer choices are correct, because some potential values of x lie outside the intervals in every other choice.

Nice question!



Why can't E be the Answer ?
User avatar
L
IanStewart
GMAT Tutor
Joined: 24 Jun 2008
Posts: 4127
Own Kudos [?]: 9599 [0]
Given Kudos: 96
 Q51  V47
Send PM
Re: If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   06 Jun 2020, 09:22
Expert Reply
anmolsd1995 wrote:
Why can't E be the Answer ?


Because the equation in the question in the question is true for, say, x = -1, so it does not need to be true that x is between 0 and 2.
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 34802
Own Kudos [?]: 876 [0]
Given Kudos: 0
Send PM
Re: If x+sqrt(x^2-4x+4)=2, then: [#permalink] If x+sqrt(x^2-4x+4)=2, then:   24 Jun 2024, 22:10
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
GMAT Club Bot
gmatclubot
Re: If x+sqrt(x^2-4x+4)=2, then: [#permalink]
24 Jun 2024, 22:10
Moderator:
Bunuel
Math Expert
95385 posts