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If g(x) = 3x + sqrt x

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diananah
 Post subject: If g(x) = 3x + sqrt x
 Post Posted: Thu Aug 04, 2011 11:05 am 
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If g(x) = 3x + sq rt of x, what is the value of g(d^2 +6d+9)?
(Chapter 5, Page 79 - In Action problem #2)

I'm struggling with the rule: sq roots have one value. In this problem, the answer key lists two solutions: one which has d+3 is greater than/= zero and the other where d+3 is less than/= zero. I'm unclear as to why this problem doesn't have just one answer - where d+3 greater than/= zero, given the general rule that when we see a sq root, we should assume it to be the positive value.
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mithunsam
 Post subject: Re: If g(x) = 3x + sqrt x
 Post Posted: Sat Aug 06, 2011 9:04 am 
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We can write g(d^2 + 6d + 9) as g((d+3)^2)

But, g(x) = 3x + sqrt(x)
Therefore, g((d+3)^2) = 3((d+3)^2) + sqrt((d+3)^2)
=> 3((d+3)^2) + (d+3) or ((d+3)^2) -(d+3)
=> 3(d^2 + 6d + 9) + (d + 3) or 3(d^2 + 6d + 9) - (d + 3)
=> 3d^2 + 18d + 27 + d + 3 or 3d^2 + 18d + 27 -d - 3
=> 3d^2 + 19d + 30 or 3d^2 + 17d + 24

It should be the solution, please verify.

Usually, we take only +ve values for the sqrt of a variable. However, in this special case, we aren't taking sqrt of a variable. We are taking sqrt of the square of a variable.

That is, sqrt((d+3)^2)

For both +ve and -ve values of (d+3), (d+3)^2 will be +ve. That means, we have to take square roots for two +ve values - one is (d+3)^2, when (d+3) is +ve and the second one is (d+3)^2, when (d+3) is -ve. So, sqrt((d+3)^2) will have two +ve solutions.

We shouldn't ignore any +ve solutions. Therefore, the problem has two solutions.
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diananah
 Post subject: Re: If g(x) = 3x + sqrt x
 Post Posted: Mon Aug 08, 2011 3:23 pm 
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That was really helpful, thanks so much!
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tim
 Post subject: Re: If g(x) = 3x + sqrt x
 Post Posted: Fri Aug 12, 2011 11:23 pm 
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cool!

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Tim Sanders
Manhattan GMAT Instructor
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sanmayved
 Post subject: Re: If g(x) = 3x + sqrt x
 Post Posted: Sun Jan 08, 2012 2:48 pm 
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I understood mithunsam's explanation but then page 98 in the same book contradicts this by saying:

Sqrt x^2 = |x|

So which one do we follow?
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tim
 Post subject: Re: If g(x) = 3x + sqrt x
 Post Posted: Sun Jan 15, 2012 4:42 pm 
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Location: Southwest Airlines, seat 21C
there is no contradiction here. sqrt(x^2) does in fact equal |x|. one way you can deal with the absolute value is to take a positive and negative version of what's inside, subject to whatever other constraints are in the problem. what both mithunsam and our book failed to do is to note that each of these answers is only valid for a certain range of possible x values..

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