document.write( "Question 329890: find all possible value of n for which the equation x^2+x-n=0 has integral solution. \n" ); document.write( "

Algebra.Com's Answer #237289 by Fombitz(32388) $\"\"$ $\"About$

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Look for solutions that look like,
\n" ); document.write( " $\"x%5E2%2Bx-n=%28x%2Ba%29%28x%2Bb%29=0\"$ where a and b are integers.
\n" ); document.write( "Using the FOIL method,
\n" ); document.write( " $\"x%5E2%2Bx-n=x%5E2%2B%28a%2Bb%29x%2Bab\"$
\n" ); document.write( "Comparing,
\n" ); document.write( "1. $\"a%2Bb=1\"$
\n" ); document.write( "2. $\"ab=-n\"$
\n" ); document.write( "From eq. 1,
\n" ); document.write( " $\"b=1-a\"$
\n" ); document.write( "Substituting into eq. 2,
\n" ); document.write( " $\"a%281-a%29=-n\"$
\n" ); document.write( " $\"highlight%28n=a%28a-1%29%29\"$ where a is any integer.
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\n" ); document.write( "Two examples:
\n" ); document.write( " $\"a=0\"$ , $\"b=1\"$ , $\"n=0\"$
\n" ); document.write( " $\"x%5E2%2Bx=x%28x%2B1%29=0\"$
\n" ); document.write( "solutions: $\"x=-a=0\"$ , $\"x=-b=-1\"$
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\n" ); document.write( " $\"a=5\"$ , $\"b=-4\"$ , $\"n=5%284%29=20\"$
\n" ); document.write( " $\"x%5E2%2Bx-20=%28x%2B5%29%28x-4%29\"$
\n" ); document.write( "solutions: $\"x=-a=-5\"$ , $\"x=-b=4\"$
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