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) About Me  (Show Source):
You can put this solution on YOUR website!
There are infinite real values of x that make sqrt%28144-sqrt%28x%29%29 a real number.
For some of those, the expression will also be an integer.
For that expression to be a real number, we need
x%3E=0 and
144-sqrt%28x%29%3E=0 --> sqrt%28x%29%3C=144 --> x%3C=144%5E2 .
In sum we need 0%3C=x%3C=144%5E2 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 sqrt%28144-sqrt%28x%29%29%3E=0 .
With x%3E=0 ,
x%3E=0 ---> sqrt%28x%29%3E=0 ---> 144-sqrt%28x%29%3C=144
0%3C=144-sqrt%28x%29%3C=144 ---> 0%3C=sqrt%28144-sqrt%28x%29%29%3C=sqrt%28144%29=12

So we need
0%3C=x%3C=144%5E2 just to have a real number,
and that will result in
0%3C=sqrt%28144-sqrt%28x%29%29%3C=12
and we want
sqrt%28144-sqrt%28x%29%29 to be an integer.
We have highlight%2813%29 integers to choose from,
and for each of those integers there is always one (and only one) real number that will make sqrt%28144-sqrt%28x%29%29 equal to that integer.
To get +0+:
sqrt%28144-sqrt%28x%29%29=0 --> 144-sqrt%28x%29=0 --> sqrt%28x%29=144 --> x=144%5E2
To get +1+:
sqrt%28144-sqrt%28x%29%29=1 --> 144-sqrt%28x%29=1%5E2=1 --> sqrt%28x%29=144-1=143 --> x=143%5E2
To get +2+:
sqrt%28144-sqrt%28x%29%29=2 --> 144-sqrt%28x%29=2%5E2=4 --> sqrt%28x%29=144-4=140 --> x=140%5E2
You can keep calculating what value of x makes sqrt%28144-sqrt%28x%29%29 equal to 3, 4, 5, and so on, all the way up to 12.
To get 12:
sqrt%28144-sqrt%28x%29%29=12 --> 144-sqrt%28x%29=12%5E2=144 --> sqrt%28x%29=144-144=0 --> x=0%5E2=0

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28144-sqrt%28x%29%29

is never negative, and x is never larger than 144²

Let n be a non-negative integer
  
sqrt%28144-sqrt%28x%29%29%22%22=%22%22n

Square both sides:

144-sqrt%28x%29%22%22=%22%22n%5E2

Isolate sqrt%28x%29 on the right:

144-n%5E2%22%22=%22%22sqrt%28x%29

%2812-n%29%2812%2Bn%29%22%22=%22%22sqrt%28x%29

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