document.write( "Question 75260: Find all positive values for k for which the following can be factored.\r
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Algebra.Com's Answer #54020 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Find all positive values for k for which the following\r\n" );
document.write( "can be factored.\r\n" );
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document.write( "x²+ x - k\r\n" );
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document.write( "It can be factored if the discriminant B²-4AC is the\r\n" );
document.write( "square of an integer. A=1, B=1, C=-k\r\n" );
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document.write( "B²-4AC = 1²-4(1)(-k) = 1+4k\r\n" );
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document.write( "We see that 1+4k is an odd number, and can be any odd\r\n" );
document.write( "perfect square.  But any odd perfect square is the\r\n" );
document.write( "square of an odd integer. \r\n" );
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document.write( "Any positive odd number can be represented by 2n+1 \r\n" );
document.write( "where n is a non-negative integer, so\r\n" );
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document.write( "1 + 4k = (2n+1)²\r\n" );
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document.write( "1 + 4k = 4n² + 4n + 1\r\n" );
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document.write( "    4k = 4n² + 4n\r\n" );
document.write( "     k = n² + n\r\n" );
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document.write( "     k = n(n+1) for any non-negative integer.\r\n" );
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document.write( "But since k can't be 0, we have to rule out n=0 \r\n" );
document.write( "and change it to\r\n" );
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document.write( "    k = n(n+1) for any POSITIVE integer n.\r\n" );
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document.write( "So k must be any term of this sequence 2, 6, 12, 20, 30, ... \r\n" );
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document.write( "Edwin

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