Advanced Quant - Workout set 5 - #43

[RaffaeleM39]

In GMAT Advanced Quant - Workout set 5 - exercise number 43, the solution has the following passage:

(a+b)/4 = (x+y)^2

sqrt{(a+b)/4} = (x+y)

I did not understand this passage.
For example, a=15, b=1, x=-1 and y=-1 would render the first equation true and the second equation false.

Moreover, below the expressions there is also written "Note that you could safely take the square root of both sides because any square is non-negative"

Can you please elaborate?

P.S. there is no Latex here, right?

[Sage Pearce-Higgins]

Please post the whole question when using these forums (although I know it's a bit cumbersome with problems with roots and exponents such as this one):

43. If a = 4x^2 + 4xy and b = 4y^2 + 4xy, which of the following is equivalent to x + y?
A) root(a + b)
B) 2 root(ab)
C) (a + b) / root2
D) 2root(a) - 2root(b)
E) root (a + b) / 2

(a+b)/4 = (x+y)^2

sqrt{(a+b)/4} = (x+y)

I did not understand this passage.
For example, a=15, b=1, x=-1 and y=-1 would render the first equation true and the second equation false.

Not necessarily. The square root of 4 is 2 or -2, so there's a solution that satisfies the second equation. However, we don't need to worry about the negative solution here (read on).

Moreover, below the expressions there is also written "Note that you could safely take the square root of both sides because any square is non-negative"

Can you please elaborate?

I think I know what's confusing. You've seen a bunch of problems in which you have to be careful to consider negative solutions to square roots. For example, if we're told that x^2 = 81 in a DS problem, then that's not sufficient information to know x (since there are two solutions). However, when a root sign is given in GMAT, we only need to consider the positive root. Take a look at this post: https://www.manhattanprep.com/gmat/foru ... ml#p117271

P.S. there is no Latex here, right?

What is "Latex"?

[RaffaeleM39]

I think I know what's confusing. You've seen a bunch of problems in which you have to be careful to consider negative solutions to square roots. For example, if we're told that x^2 = 81 in a DS problem, then that's not sufficient information to know x (since there are two solutions). However, when a root sign is given in GMAT, we only need to consider the positive root. Take a look at this post: https://www.manhattanprep.com/gmat/foru ... ml#p117271

But here the question stem does not contain any root sign. Nor we can infer positive/negative.
I mean, if I write that the side of a square is sqrt(4), then I obviously mean the positive solution (the side is 2, not -2, because a side must have a positive length).
But here the question stem does not contain any root sign. However:

The square root of 4 is 2 or -2, so there's a solution that satisfies the second equation. However, we don't need to worry about the negative solution here (read on).

Here you are theoretically right, but as Ron said in the post you cited, mathematical functions must be unambiguous.
In fact, if you look up square root in Wikipedia it reports what I have learned in school:

Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by √a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by √9 = 3, because 32 = 3 • 3 = 9 and 3 is nonnegative.

So the square root symbol only denotes the positive solution, and not the negative solution. That is to say, a mathematical function must output a unique value.

In your example, how would you know how to pair the negative and positive sign?

So in the example in question, this is what I would have done:

(a+b)/4 = (x+y)^2 => sqrt{(a+b)/4} = |x+y| = (x+y) or (-x-y)

So I am a bit confused: when we should take into consideration the positive/negative sign and where not?

What is "Latex"?

Latex is a typeset system. It is used, among other purposes, in some mathematical forums to render equations correctly (instead of seeing sqrt(2)/2, you see the actual square root sign and the fraction)

[Sage Pearce-Higgins]

Ron' distinction between square roots that are given, and those you need to apply is a good one. Since the square root sign is in the answer choice, you can take that as being part of the problem (i.e. it's a given square root, rather than one that you need to apply yourself). You could see that, working back from answer E gives equivalence. In any case, the problem would make sense if we considered the two root solutions.

That said, GMAT is pretty good at avoiding any ambiguity. I'm happy to accept that our Advanced Quant workouts may not be written to the same high standards as real GMAT problems and therefore take your cues on this issue from OG or GMAT Prep problems. Take for example problems 350 and 401. They should give you the best idea of what to expect in this area in the GMAT.

No Latex here unfortunately.