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

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If x+sqrt(x^2-4x+4)=2, then:  [#permalink]

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New post Updated on: 27 Mar 2019, 12:52
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Question Stats:

19% (01:35) correct 81% (01:56) wrong based on 59 sessions

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GMATH practice exercise (Quant Class 12)

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

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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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.
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If x+sqrt(x^2-4x+4)=2, then:  [#permalink]

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New post 27 Mar 2019, 12:56
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!
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Re: If x+sqrt(x^2-4x+4)=2, then:  [#permalink]

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New post 27 Mar 2019, 13:10
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!
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Re: If x+sqrt(x^2-4x+4)=2, then:  [#permalink]

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New post 27 Mar 2019, 13:30
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fskilnik wrote:
GMATH practice exercise (Quant Class 12)

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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.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: If x+sqrt(x^2-4x+4)=2, then:   [#permalink] 27 Mar 2019, 13:30
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