SOLUTION: For how many real numbers x, is { { { sqrt ( 144 - sqrt ( x ) ) } } } an integer?

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Question 823887: For how many real numbers x, is { { { sqrt ( 144 - sqrt ( x ) ) } } } an integer?
Found 2 solutions by KMST, Edwin McCravy:
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
There are infinite real values of that make a real number.
For some of those, the expression will also be an integer.
For that expression to be a real number, we need
and
--> --> .
In sum we need just to end up with a real number.
That imposes a restriction but we would still have an infinite number of options.
We know that .
With ,
---> --->
--->

So we need
just to have a real number,
and that will result in

and we want
to be an integer.
We have integers to choose from,
and for each of those integers there is always one (and only one) real number that will make equal to that integer.
To get :
--> --> -->
To get :
--> --> -->
To get :
--> --> -->
You can keep calculating what value of makes equal to 3, 4, 5, and so on, all the way up to 12.
To get :
--> --> -->
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!


is never negative, and x is never larger than 144²

Let n be a non-negative integer
  


Square both sides:



Isolate  on the right:





Since the right side is never negative, n can only take on 
the 13 non-negative integers from 0 through 12.

(12-n)²(12+n)² = x, for n = 0,1,2,...,12 

Answer: there are 13 values x can take on. 

Edwin
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