Someone posted this tough question:

Source: Gmat Prep, mba.com, Test II

This question is a little challenging to post, but hopefully it is clear. I am looking for the fastest and most efficient way to solve this problem. I used estimation when I did it, and that left 2 answers that were very close. So, I had to go back and use more accurate estimation and got the answer (this took a little bit longer than I think this question deserves, so I was hoping for some tips!). Thanks!

(rt(9 + rt80) + rt(9 - rt80))^2 =

a. 1
b. 9 - 4.rt5
c. 18 - 4.rt5
d. 18
e. 20


Stacey replied this:

Tough question! If you're going for a top score, make sure you know the three common quadratic "perfect square" equations and how to use them with a weird variation like this. This one's complicated even with the "shortcut."

This starts as a variation of (a+b)^2 = a^2 + 2ab + b^2.
so:
[rt(9 + rt80) + rt(9 - rt80)]^2 =
[rt(9 + rt80)]^2 + [(2)(rt(9 + rt80)rt(9 - rt80)] + [rt(9 - rt80)]^2 Square roots on first and third terms cancel out:
(9 + rt80) + [(2)(rt(9 + rt80)rt(9 - rt80)] + (9 - rt80)
and middle term is now a variation of (a+b)(a-b) = a^2 - b^2:
(9 + rt80) + [(2))rt{(9 + rt80)(9 - rt80)}] + (9 - rt80)
(9 + rt80) + [(2)(81-80)] + (9 - rt80)
9 + 2 + 9 = 20


I got lost here:

[rt(9 + rt80)]^2 + [(2)(rt(9 + rt80)rt(9 - rt80)] + [rt(9 - rt80)]^2 Square roots on first and third terms cancel out:
(9 + rt80) + [(2)(rt(9 + rt80)rt(9 - rt80)] + (9 - rt80)

How do you do that?. From this: [rt(9 + rt80)]^2, how do you get this: (9 + rt80),
shouldn´t it be (9rt)^2+[80(rt)^2]^2+18(80)(rt)^2

And later in that post the guy that post it asked why another solution was wrong but he didn´t get any response. And I´m asking the same thing because it made sens that answer for me also. The thing is, if you have this:

(rt(9 + rt80) + rt(9 - rt80))^2 = You can first solve what is inside and you´ll get:
(2(9)rt)^2=324(rt)^2.

It seems much easier and must be wrong because there is no such option in the 5 posible answers.But the strange thing is that this is exactly what I got solving all the mess I´have put above in the part where I didn´t understnad Stacey´s explanation. Where I´m getting lost and confuse. Can anyone help me here?

Thank you

From what you wrote, it might be the case that you're reading the "rt" as variables - but "rt" is an abbreviation for "square root" (since we can't show square root signs on the forum). If that is the case, go back and take a look with the understanding that "rt" doesn't represent variables but the mathematical expression for square root. I use the abbreviation "SQRT" below to clarify, just in case.

If you have, for example (SQRTx)^2, then the square root and the square just cancel each other out and you're left with x. (Try writing this on paper with normal symbols - it's hard to look at it in this format.)

This is just a more complicated term:
[rt(9 + rt80)]^2
but it's the exact same thing - the SQRT and the ^2 cancel each other out and you're left only with what's in the parentheses: (9 + rt80). Again, write this out on paper so you can really follow what's going on. Think of that entire term (9 + rt80) as "x," if that helps.

I'm not sure what the specific second question was (that someone else asked), so not sure how to address that. But if you do the above math correctly, you should get to the right answer.

Does that make more sense?

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

I want to make sure I did the calculation for the middle part correctly. Could someone please show me how one went about solving the middle part of this explanation. I cut and paste the portion of the problem below: the portion is highlighted.


(9 + rt80) + [(2))rt{(9 + rt80)(9 - rt80)}] + (9 - rt80)
(9 + rt80) + [(2)(81-80)] + (9 - rt80)
9 + 2 + 9 = 20

the question is how did one get 81-80.

your response is greatly appreciated.

Hi Guest, let's focus on the underlined part first: i.e.
(9 + rt80)(9 - rt80)

realise that this is in the form of (a+b)*(a-b), where a=9 and b=rt80
this will give you a-squared minus b-squared
i.e. 9-squared minus rt80-squared
i.e. 81 minus 80
i.e. 1

and 2 times rt1 is 2.

the whole expression, therefore, simplifies to
(9 + rt80) + 2 + (9 - rt80)
i.e. 9 + 2 + 9 (since +/- rt80 will cancel each other)
i.e. 20

Hope this helps.

i agree with the answer of 20 how ever during the last step rt(1) could be both +ve and -ve....hence the answer cud eventually be either 18+2 = 20 or 18-2 = 16.any suggestions ?

nope, the square root is defined in exactly one way: namely, the positive square root. if you see √x, that refers only to the positive square root of x, not both possibilities.

you have to make sure that you differentiate between EQUATIONS and EXPRESSIONS. specifically:

EQUATIONS, which feature an equals sign and a variable to be SOLVED FOR, can, and often do, have more than one solution.
in particular, if you have the equation x^2 = 9, there are two solutions: x = 3 and x = -3. these are both solutions because both of them, when plugged in, make the equation true.

however,
EXPRESSIONS, which feature operator symbols (such as the √ symbol), always have only ONE VALUE.
so if you see √9, that means only 3. it does NOT mean either 3 or -3, because operator symbols can only give one answer.

hope that helps.

here's the above post, edited to include square root signs, because we have consummate style.

this method is nothing short of brilliant - but you have to think of it!
to the poster who posted this: if you can think to do this sort of thing within two minutes, you are amazing at this stuff.