document.write( "Question 1102396: Solve equation
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Algebra.Com's Answer #717212 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Tutor Josmiceli gave these four answers:\r\n" );
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document.write( "\"+x+=+sqrt%28+3+%2B+2%2Asqrt%282%29+%29+\"\r\n" );
document.write( "\"+x+=+-sqrt%28+3+%2B+2%2Asqrt%282%29+%29+\"\r\n" );
document.write( "\"+x+=+sqrt%28+3+-+2%2Asqrt%282%29+%29+\"\r\n" );
document.write( "\"+x+=+-sqrt%28+3+-+2%2Asqrt%282%29+%29+\"\r\n" );
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document.write( "His work was correct up to that point.  But he didn't finish.\r\n" );
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document.write( "1. He didn't rule out the two extraneous answers.\r\n" );
document.write( "2. He didn't simplify the two answers which are solutions.\r\n" );
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document.write( "His second and fourth answers are negative which are not in\r\n" );
document.write( "the domain of the log function.  So the only solutions are\r\n" );
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document.write( "\"+x+\"\"%22%22=%22%22\"\"+sqrt%28+3+%2B+2%2Asqrt%282%29+%29+\"\r\n" );
document.write( "\"+x\"\"%22%22=%22%22\"\"sqrt%28+3+-+2%2Asqrt%282%29+%29+\"\r\n" );
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document.write( "or \"+x\"\"%22%22=%22%22\"\"sqrt%283+%2B-+2%2Asqrt%282%29+%29+\"\r\n" );
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document.write( "However they are not in simplest radical form, as they\r\n" );
document.write( "contain a square root inside a square root.\r\n" );
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document.write( "We set  \r\n" );
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document.write( "\"sqrt%283+%2B-+2%2Asqrt%282%29+%29\"\"%22%22=%22%22\"\"u+%2B-+sqrt%28v%29\" \r\n" );
document.write( "where u and v are rational.\r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "\"3+%2B-+2%2Asqrt%282%29\"\"%22%22=%22%22\"\"u%5E2+%2B-+2u%2Asqrt%28v%29+%2B+v\"\r\n" );
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document.write( "We equate the terms without square roots and\r\n" );
document.write( "equate terms with square roots, and we have this\r\n" );
document.write( "system:\r\n" );
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document.write( "\"system%283=u%5E2%2Bv%2C%22%22+%2B-+2%2Asqrt%282%29=%22%22+%2B-+2u%2Asqrt%28v%29%29\"\r\n" );
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document.write( "Divide the second equation through by \"%22%22+%2B-+2\"\r\n" );
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document.write( "\"sqrt%282%29\"\"%22%22=%22%22\"\"u%2Asqrt%28v%29\"\r\n" );
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document.write( "Squaring both sides:\r\n" );
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document.write( "\"2\"\"%22%22=%22%22\"\"u%5E2v\"\r\n" );
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document.write( "\"2%2Fu%5E2\"\"%22%22=%22%22\"\"v\"\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "\"3\"\"%22%22=%22%22\"\"u%5E2%2Bv\"\r\n" );
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document.write( "\"3\"\"%22%22=%22%22\"\"u%5E2%2B2%2Fu%5E2\"\r\n" );
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document.write( "Multiply through by u2\r\n" );
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document.write( "\"3u%5E2\"\"%22%22=%22%22\"\"u%5E4%2B2\"\r\n" );
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document.write( "\"0\"\"%22%22=%22%22\"\"u%5E4-3u%5E2%2B2\"\r\n" );
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document.write( "\"0\"\"%22%22=%22%22\"\"%28u%5E2-1%29%28u%5E2-2%29\"\r\n" );
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document.write( "\"u%5E2-1=0\";  \"u%5E2-2=0\"\r\n" );
document.write( "\"u%5E2=1\";  \"u%5E2=2\"\r\n" );
document.write( "\"u=%22%22+%2B-+1\";   \"u=%22%22+%2B-+sqrt%282%29\"\r\n" );
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document.write( "Since we assumed u and v are rational, we discard\r\n" );
document.write( "the second answers for u.\r\n" );
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document.write( "Therefore:\r\n" );
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document.write( "\"sqrt%283+%2B-+2%2Asqrt%282%29+%29\"\"%22%22=%22%22\"\"u+%2B-+sqrt%28v%29\"\"%22%22=%22%22\"\"%22%22+%2B-+1+%2B-+sqrt%282%29\"\r\n" );
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document.write( "However we must discard the negative values \"1-sqrt%282%29\" and\r\n" );
document.write( "\"-1-sqrt%282%29\" as extraneous.\r\n" );
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document.write( "So the only two solutions in simplified form are\r\n" );
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document.write( "\"%22%22+%2B-+1+%2B+sqrt%282%29\"\r\n" );
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document.write( "Edwin
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