document.write( "Question 1196157: Find the modulus and argument of √(1+2i)/(1-2i) \n" ); document.write( "
Algebra.Com's Answer #828898 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Did you mean this: \"sqrt%281%2B2i%29%2F%281-2i%29\" or this: \"sqrt%28%281%2B2i%29%2F%281-2i%29%29\"?\r\n" );
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document.write( "I'll assume the second\r\n" );
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document.write( "\"sqrt%28%281%2B2i%29%2F%281-2i%29%29\"\r\n" );
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document.write( "\"sqrt%28%28%281%2B2i%29%281%2B2i%29%29%2F%28%281-2i%29%281%2B2i%29%29%29\"\r\n" );
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document.write( "\"sqrt%28%281%2B4i%2B4i%5E2%29%2F%281-4i%5E2%29%29\"\r\n" );
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document.write( "\"sqrt%28%281%2B4i%2B4%28-1%29%29%2F%281-4%28-1%29%29%29\"\r\n" );
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document.write( "\"sqrt%28%281%2B4i-4%29%2F%281%2B4%29%29\"\r\n" );
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document.write( "\"sqrt%28%28-3%2B4i%29%2F%285%29%29\"\r\n" );
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document.write( "\"sqrt%281%2F5%29sqrt%28-3%2B4i%29\"\r\n" );
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document.write( "\"sqrt%28expr%281%2F5%29%2Aexpr%285%2F5%29%29sqrt%28-3%2B4i%29\"\r\n" );
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document.write( "\"expr%28sqrt%285%29%2F5%29sqrt%28-3%2B4i%29\"\r\n" );
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document.write( "let\r\n" );
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document.write( "\"sqrt%28-3%2B4i%29\"\"%22%22=%22%22\"\"a%2Bbi\"\r\n" );
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document.write( "\"-3%2B4i\"\"%22%22=%22%22\"\"a%5E2%2B2abi%2Bb%5E2i%5E2\"\r\n" );
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document.write( "\"-3%2B4i\"\"%22%22=%22%22\"\"a%5E2%2B2abi%2Bb%5E2%28-1%29\"\r\n" );
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document.write( "\"-3%2B4i\"\"%22%22=%22%22\"\"a%5E2%2B2abi-b%5E2\"\r\n" );
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document.write( "Setting real parts equal and imaginary parts equal.\r\n" );
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document.write( "\"system%28a%5E2-b%5E2=-3%2C+2ab=4%29\"\r\n" );
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document.write( "\"system%28a%5E2-b%5E2=-3%2C+ab=2%29\"\r\n" );
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document.write( "Solve that system and get a = ±1, b = ±2\r\n" );
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document.write( "So there are two solutions to the system: ±1±2i\r\n" );
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document.write( "In either case the modulus is \"%28%22%22%2B-1%29%5E2%2B%28%22%22%2B-2%29%5E2\"\"%22%22=%22%22\"\"sqrt%281%2B4%29\"\"%22%22=%22%22\"\"sqrt%285%29\"\r\n" );
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document.write( "But the modulus must be multiplied by the factor \"sqrt%285%29%2F5\"\r\n" );
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document.write( "\"modulus\"\"%22%22=%22%22\"\"sqrt%285%29%2F5\"\"%22%22%2A%22%22\"\"sqrt%285%29\"}}}\"\"=\"\"}}}\"5%2F5\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "In the case of 1+2i, which is in the first quadrant with \r\n" );
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document.write( "\"tan%28argument%29\"\"%22%22=%22%22\"\"2%2F1\"\"%22%22=%22%22\"\"2\"\r\n" );
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document.write( "\"argument\"\"%22%22=%22%22\"\"63.43%5Eo\"\r\n" );
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document.write( "In the case of -1-2i, which is in the third quadrant with \r\n" );
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document.write( "\"tan%28argument%29\"\"%22%22=%22%22\"\"%28-2%29%2F%28-1%29\"\"%22%22=%22%22\"\"2\"\r\n" );
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document.write( "\"argument\"\"%22%22=%22%22\"\"243.43%5Eo\"\r\n" );
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document.write( "Edwin
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