Negative solutions to square roots?

[tgilham]

I'm confused by the apparently conflicting rules on GMAT regarding this topic. Why is it that Sq root of 25 cannot equal -5, but if it is part of an equation where you square root by sides, it can do?

Just wondering whether I'm missing a step in the logic?

Thanks

Tom

[jnelson0612]

Yes, this is a confusing aspect on the GMAT. Just remember the following:

1) If I have x^2=25, then x could be 5 or -5. Either of those numbers squared equal 25.

2) If you are asked for the square root of 25, give ONLY the positive value, 5. Negative square roots are called imaginary numbers and the GMAT has chosen not to deal with those.

Hope this clarifies things!

[khandelwal.ab]

Hi Jamie,

Thanks for your response. Even I have been struggling with this concept.

Your above response clarifies it, but I have a followup question.

What if the root contains an unknown?

for instance what would be the solution for root of (d+3)^2 as per rule number 2 you mentioned above, it should be just (D+3), but i have seen such examples at many places (including Mgmat books), where we have two solutions for such expressions -(D+3) and (D+3).

It will be great if you could provide us with some clarity here..

Thanks in advance!

[RonPurewal]

this is not anything "gmat-specific". it's a mathematical convention followed worldwide, with no exceptions anywhere. and, no, there's no contradiction.

the point is this: symbols / operators have to mean ONE thing. they can't be ambiguous.
so, the problem that lies before us is to define the "√" sign IN ONE WAY, just as in the case of any other symbol.
there are two choices:
1/ we could make "√n" stand for the negative value;
2/ we could make "√n" stand for the positive value.
i don't think i need to explain why #1 would be absurd here, so, accordingly, the convention is #2.

also, it's a darned good thing that symbols only mean one thing.
for instance, how long is the diagonal of a square with 1-inch sides?
"√2", you say.
you're right.
if the "√" symbol were allowed to be either positive or negative, then it would actually be impossible to answer this simple question.

--

when you have an EQUATION, though -- like "x^2 = 9" -- this issue isn't in play, because there is no "√" symbol that you are trying to define uniquely. "solving" an equation means finding ALL numbers that make the statement true, so there will be two solutions to equations like these.

in any case, it's a misrepresentation to write that this is a "GMAT convention", as that would imply the possibility of finding some remote mathematical outpost, somewhere, where "√n" is allowed to stand for a negative value. not true -- the same convention is followed everywhere, for the reasons outlined above.

[khandelwal.ab]

Thanks a lot for the explanation Ron.

Just to confirm that I got it right, I'll come back to the example I mentioned in my earlier post.

What will be the possible solution/s for √(d+3)^2 ?

Since, this is not an equation would there be just one solution i.e. D+3?

I am sorry if this is a dim question, but I want to be completely sure about this concept.

Thanks a lot for your time.

[jnelson0612]

Thanks a lot for the explanation Ron.

Just to confirm that I got it right, I'll come back to the example I mentioned in my earlier post.

What will be the possible solution/s for √(d+3)^2 ?

Since, this is not an equation would there be just one solution i.e. D+3?

I am sorry if this is a dim question, but I want to be completely sure about this concept.

Thanks a lot for your time.

The tricky thing here is what is d? If d=-4, then the ultimate value after squaring and taking the square root is 1. However, (d+3) is (-4 +3) or -1.

On the other hand, if d=3, then the ultimate value is 6, which is equal to (d+3).

You are making this more complicated by adding a variable which is then combined with another integer and squared. I could say that the square root of this problem will always be |d+3|.

[msavetsila]

Yes, this is a confusing aspect on the GMAT. Just remember the following:

1) If I have x^2=25, then x could be 5 or -5. Either of those numbers squared equal 25.

2) If you are asked for the square root of 25, give ONLY the positive value, 5. Negative square roots are called imaginary numbers and the GMAT has chosen not to deal with those.

Hope this clarifies things!

Hope this is not out of scope but what if odd root like root 3 of -27....
would that be -3?

Thank you for your time

[RonPurewal]

Hope this is not out of scope but what if odd root like root 3 of -27....
would that be -3?

Thank you for your time

yes, odd roots keep their signs. the cube root of -27 is -3, and the cube root of +27 is +3.

there's no issue at all with odd roots, because the correspondence is one-to-one; the operation is completely reversible. (by contrast, squaring is not reversible. if i square 3 and -3, i get 9 both times, and it's impossible to tell which value i started with.)

i am moving this thread to the general math folder, where it belongs. mind the folders, folks -- thanks.

[arpitsharms]

Yes, this is a confusing aspect on the GMAT. Just remember the following:

1) If I have x^2=25, then x could be 5 or -5. Either of those numbers squared equal 25.

2) If you are asked for the square root of 25, give ONLY the positive value, 5. Negative square roots are called imaginary numbers and the GMAT has chosen not to deal with those.

Hope this clarifies things!

Hi Jamie
I am still pretty confused about this.Hope you can help me out.
An Imaginary no. would be sqrt (-25).Now neither 5 nor -5 can be its solution.In fact no real no. can be its solution.
But as you pointed out that if x^2=25,then x could be either 5 or -5
Therefore sqrt 25 can have two values either 5 or -5.
Why is the negative value(-5) discarded?

[jnelson0612]

Hi Jamie
I am still pretty confused about this.Hope you can help me out.
An Imaginary no. would be sqrt (-25).Now neither 5 nor -5 can be its solution.In fact no real no. can be its solution.
But as you pointed out that if x^2=25,then x could be either 5 or -5
Therefore sqrt 25 can have two values either 5 or -5.
Why is the negative value(-5) discarded?

It's really simple:
1) If something is SQUARED and produces a positive integer, that number could be positive or negative. x^2 = 25 so 5 or -5 squared are 25.

2) If something is SQUARE ROOTED, then the value is only positive. Square root of 25 is only 5.

Just flashcard and memorize this, and then don't worry about it! There is so much to worry about with the GMAT without taking on any more worries than you need to! :-)

[arpitsharms]

Ok I will.And thanks for the help.

[RonPurewal]

imaginary numbers are irrelevant to this exam. only real numbers.

[RonPurewal]

the same principle holds true for imaginary numbers anyway; the symbol "√–25" is uniquely defined as 5i, not as ±5i. again, though, doesn't matter for this exam.

[smunir999]

Hello,
How about if you have underroot (x^2). then do you get a +_ x, or just positive x?

[RonPurewal]

assuming "underroot" means "√":

• if x is negative, then √(x^2) is the same size, but positive.
for instance, if x = -4, then √(x^2) = 4.

• if x is zero or positive, then √(x^2) is exactly the same as x.
for instance, if x = 11, then √(x^2) = 11.

so, we have:
• if x is negative, the sign flips.
• if x is positive or zero, the sign stays the same.

the above happens to be the exact definition of absolute value. so, we have √(x^2) = |x|.