document.write( "Question 823887: For how many real numbers x, is { { { sqrt ( 144 - sqrt ( x ) ) } } } an integer? \n" ); document.write( "

Algebra.Com's Answer #495908 by KMST(5328) $\"\"$ $\"About$

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There are infinite real values of $\"x\"$ that make $\"sqrt%28144-sqrt%28x%29%29\"$ a real number.
\n" ); document.write( "For some of those, the expression will also be an integer.
\n" ); document.write( "For that expression to be a real number, we need
\n" ); document.write( " $\"x%3E=0\"$ and
\n" ); document.write( " $\"144-sqrt%28x%29%3E=0\"$ --> $\"sqrt%28x%29%3C=144\"$ --> $\"x%3C=144%5E2\"$ .
\n" ); document.write( "In sum we need $\"0%3C=x%3C=144%5E2\"$ just to end up with a real number.
\n" ); document.write( "That imposes a restriction but we would still have an infinite number of options.\r
\n" ); document.write( "\n" ); document.write( "We know that $\"sqrt%28144-sqrt%28x%29%29%3E=0\"$ .
\n" ); document.write( "With $\"x%3E=0\"$ ,
\n" ); document.write( " $\"x%3E=0\"$ ---> $\"sqrt%28x%29%3E=0\"$ ---> $\"144-sqrt%28x%29%3C=144\"$
\n" ); document.write( " $\"0%3C=144-sqrt%28x%29%3C=144\"$ ---> $\"0%3C=sqrt%28144-sqrt%28x%29%29%3C=sqrt%28144%29=12\"$
\n" ); document.write( "
\n" ); document.write( "So we need
\n" ); document.write( " $\"0%3C=x%3C=144%5E2\"$ just to have a real number,
\n" ); document.write( "and that will result in
\n" ); document.write( " $\"0%3C=sqrt%28144-sqrt%28x%29%29%3C=12\"$
\n" ); document.write( "and we want
\n" ); document.write( " $\"sqrt%28144-sqrt%28x%29%29\"$ to be an integer.
\n" ); document.write( "We have $\"highlight%2813%29\"$ integers to choose from,
\n" ); document.write( "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.
\n" ); document.write( "To get $\"+0+\"$ :
\n" ); document.write( " $\"sqrt%28144-sqrt%28x%29%29=0\"$ --> $\"144-sqrt%28x%29=0\"$ --> $\"sqrt%28x%29=144\"$ --> $\"x=144%5E2\"$
\n" ); document.write( "To get $\"+1+\"$ :
\n" ); document.write( " $\"sqrt%28144-sqrt%28x%29%29=1\"$ --> $\"144-sqrt%28x%29=1%5E2=1\"$ --> $\"sqrt%28x%29=144-1=143\"$ --> $\"x=143%5E2\"$
\n" ); document.write( "To get $\"+2+\"$ :
\n" ); document.write( " $\"sqrt%28144-sqrt%28x%29%29=2\"$ --> $\"144-sqrt%28x%29=2%5E2=4\"$ --> $\"sqrt%28x%29=144-4=140\"$ --> $\"x=140%5E2\"$
\n" ); document.write( "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.
\n" ); document.write( "To get $\"12\"$ :
\n" ); document.write( " $\"sqrt%28144-sqrt%28x%29%29=12\"$ --> $\"144-sqrt%28x%29=12%5E2=144\"$ --> $\"sqrt%28x%29=144-144=0\"$ --> $\"x=0%5E2=0\"$ \n" ); document.write( "

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