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

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Wed Mar 27, 2019 11:57 am

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GMATH practice exercise (Quant Class 12)

Answer: [spoiler]______ (A)__[/spoiler]

Hint: [spoiler]The alternative choices are tricky... the official answer is NOT wrong... and you can find it in 3 SECONDS, using LOGIC only!
Obs.: someone could argue that GMAT asks you to find the BEST answer, therefore the 3-seconds approach may be considered "risky"... [/spoiler]

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

Source: Beat The GMAT — Problem Solving | Wed Mar 27, 2019 11:57 am

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

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

by fskilnik@GMATH » Wed Mar 27, 2019 12:39 pm

fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 12)

Answer: [spoiler]______ (A)__[/spoiler]

Important: Have you realized (looking at the alternative choices carefully) that the answer MUST BE (A)?
Reason: if (B), (C), (D), or (E) is true, then (A) is also true!

On the other hand, the GMAT asks you to find the BEST answer, therefore this approach could be considered "risky"... let me explain why:

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 choice (A) is also true.

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... now let´s go to "the official solution":

$$?\,\,\,:\,\,\,\,x\,\,{\rm{must}}\,\,{\rm{be}}\,\,\, \ldots $$
$$x + \sqrt {{x^2} - 4x + 4} \,\,\, = \,\,\,2\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\sqrt {{{\left( {x - 2} \right)}^2}} = 2 - x$$
$$\,\,\,\, \Leftrightarrow \,\,\,\,\,\left| {x - 2} \right| = - \left( {x - 2} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x - 2 \le 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \le 2$$
$$x \le 2\,\,\,\, \Rightarrow \,\,\,\left( A \right)\,\,{\rm{is}}\,\,{\rm{true}}\,\,\,\,\left[ {{\rm{and}}\,\,\left( B \right),\left( C \right),\left( D \right),\left( E \right)\,\,{\rm{are}}\,\,{\rm{false}}} \right]$$

The correct answer is (A).

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

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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