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squares versus square roots

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by gmatdestroyer1 » Sat May 03, 2014 3:56 pm
I know it's a rule that when x^2 = 81, the solution is x = 9 or -9, and when x = sqrt(81), the answer is just positive 9. So, I just take this as a given on the GMAT.

But I'm still wondering... why is this true?? It doesn't seem to make sense.
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by Brent@GMATPrepNow » Sat May 03, 2014 4:42 pm
This all has to do with agreed-upon notation. From the Official Guide:

A square root of a number n is a number that, when squared, is equal to n. Every positive number n has two square roots, one positive and the other negative, but √n denotes the positive number whose square is n. For example, √9 denotes 3.


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by dabral » Fri May 23, 2014 10:22 am
It is an agreed upon convention, which means √81=9. If we have x^2=81, then we write x = +√81= 9 or x= -√81= -9.

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by sukriti2hats » Sat May 31, 2014 3:56 am
It usually depends on what is being asked in the question. For example if the question asks you to find weight or distance or something that you know can certainly not be negative then you just neglect the negative root of that quadratic equation. Otherwise, there always exist 2 roots of a quadratic equation and if the question is typically about finding the roots then just giving the positive root will suffice.
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by [email protected] » Sat May 31, 2014 9:07 am
Hi All,

There's actually a rather simple way to deal with this issue. Focus on what the prompt GIVES YOU to work with.

If you're given a square root sign, then you are responsible for the POSITIVE answer ONLY.

If you're given a squared term (e.g. X^2 = 25), then you are responsible for BOTH the POSITIVE and NEGATIVE answers.

By extension, if you're given a quadratic (e.g. X^2 +2X - 3), then you're responsible for BOTH answers (regardless of whether they're positive or negative).

In questions involving logical restrictions (geometry, etc.), there's no such thing as a "negative side length", which is why the Pythagorean Theorem is concerned with just the positive results.

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