document.write( "Question 1189716: Given that x and y are integers and $\"+sqrt%28+2+sqrt%28+10+%29+%2B11+%29+=+x%2B+sqrt%28+x%2By+%29+\"$ , find x-y \n" ); document.write( "

Algebra.Com's Answer #821143 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Given that x and y are integers and $\"sqrt%282%2Asqrt%2810%29%2B11%29\"$ = x + $\"sqrt%28x%2By%29\"$ , find x-y.
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document.write( "Your starting equality is\r\n" );
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document.write( "     = x + .                 (1)\r\n" );
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document.write( "Square both sides. You will get\r\n" );
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document.write( "     = x^2 + 2x*sqrt(x+y) + (x+y),\r\n" );
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document.write( "or, equivalently,\r\n" );
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document.write( "     = x^2 + (x+y) + 2x*sqrt(x+y).    (2)\r\n" );
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document.write( "Since x and y are integer, we conclude\r\n" );
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document.write( "    11 = x^2 + x + y           (3)\r\n" );
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document.write( "    sqrt(10) = x*sqrt(x+y)     (4)\r\n" );
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document.write( "Square (4) again. You will get\r\n" );
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document.write( "    10 = x^2*(x + y).              (5)\r\n" );
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document.write( "10 = 1*10 = 2*5. Based on uniqueness of decomposition of the number 10 into the product of primes,\r\n" );
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document.write( "we conclude that equality (5) has a unique solution x= 1 and y= 9 in integer numbers.\r\n" );
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document.write( "Notice that equality  (3) is valid, too, with x= 1 and y= 9.\r\n" );
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document.write( "So, this pair (x,y) = (1,9) is the unique solution for equation (2), and, hence, for equation (1).\r\n" );
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document.write( "ANSWER.  x - y = 1 - 9 = -8.\r\n" );
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