document.write( "Question 1208227: If x² = x + 1 and y² = y + 1.
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Algebra.Com's Answer #846533 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "An easier and more elegant way is to use\r\n" );
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document.write( "\"x%5E2+=+x+%2B+1\"\r\n" );
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document.write( "as a recurrence formula for higher powers of x\r\n" );
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document.write( "\"x%5E3=x%28x%5E2%29=x%28x%2B1%29=x%5E2%2Bx=%28x%2B1%29%2Bx=2x%2B1\"\r\n" );
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document.write( "\"x%5E4=x%28x%5E3%29=x%282x%2B1%29=2x%5E2%2Bx=2%28x%2B1%29%2Bx=3x%2B2\"\r\n" );
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document.write( "\"x%5E5=x%28x%5E4%29=x%283x%2B2%29=3x%5E2%2B2x=3%28x%2B1%29%2B2x=5x%2B3\"\r\n" );
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document.write( "So \"x%5E5=5x%2B3\" and \"y%5E5=5y%2B3\"\r\n" );
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document.write( "Since \"x%5E2+=+x+%2B+1\" and \"y%5E2+=+y+%2B+1\"  both have solutions \r\n" );
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document.write( "\"1%2F2+%2B-+sqrt%285%29%2F2+\" \r\n" );
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document.write( "\"x%5E5%2By%5E5=5x%2B5y%2B6=5%28x%2By%29%2B6\"\r\n" );
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document.write( "There are 4 possibilities for x+y\r\n" );
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document.write( "\"x%2By%29\"\"%22%22=%22%22\"\"%281%2F2+%2B+sqrt%285%29%2F2%29%2B%281%2F2+%2B+sqrt%285%29%2F2%29=1%2Bsqrt%285%29\" \r\n" );
document.write( "\"x%2By%29\"\"%22%22=%22%22\"\"%281%2F2+%2B+sqrt%285%29%2F2%29%2B%281%2F2+-+sqrt%285%29%2F2%29=1\" \r\n" );
document.write( "\"x%2By%29\"\"%22%22=%22%22\"\"%281%2F2+-+sqrt%285%29%2F2%29%2B%281%2F2+%2B+sqrt%285%29%2F2%29=1\" \r\n" );
document.write( "\"x%2By%29\"\"%22%22=%22%22\"\"%281%2F2+-+sqrt%285%29%2F2%29%2B%281%2F2+-+sqrt%285%29%2F2%29=1%2Bsqrt%285%29\" \r\n" );
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document.write( "Substituting in:\r\n" );
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document.write( "\"x%5E5%2By%5E5=5%28x%2By%29%2B6\"\r\n" );
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document.write( "\"x%5E5%2By%5E5=5%281%2Bsqrt%285%29%29%2B6=11%2Bsqrt%285%29\"\r\n" );
document.write( "\"x%5E5%2By%5E5=5%281%29%2B6=11\"\r\n" );
document.write( "\"x%5E5%2By%5E5=5%281%2Bsqrt%285%29%29%2B6=11-sqrt%285%29\"\r\n" );
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document.write( "Moral: If you work at a problem long enough you will find the simplest\r\n" );
document.write( "solution.  It's much easier when you have someone else's longer solution to\r\n" );
document.write( "simplify.  And nobody is STUPID for failure to use the shortest possible\r\n" );
document.write( "solution first.\r\n" );
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document.write( "Edwin
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