document.write( "Question 1083004: Heya, I'd be really greatful if you could solve this question
\n" ); document.write( "from calculus, complex numbers for me.
\n" ); document.write( "Find the complex number u = x + yi, where x and y are real
\n" ); document.write( "numbers, so that u^2 = -5 + 12i. Then solve the equation
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Algebra.Com's Answer #697009 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
Heya, I'd be really greatful if you could solve this question
\n" ); document.write( "from calculus, complex numbers for me.
\n" ); document.write( "Find the complex number u = x + yi, where x and y are real
\n" ); document.write( "numbers, so that u^2 = -5 + 12i. Then solve the equation
\n" ); document.write( "z^2+zi+(1-3i)=0\r
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document.write( "When we solve this quadratic equation:\r\n" );
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document.write( "\"z%5E2%2Bzi%2B%281-3i%29=0\"\r\n" );
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document.write( "by the quadratic formula and simplify, we get this:\r\n" );
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document.write( "\"z+=+%28-i+%2B-+sqrt%28-5%2B12i%29+%29%2F2+\" \r\n" );
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document.write( "To simplify further we need to find a simpler form for\r\n" );
document.write( "the square root, say u = x + yi such that\r\n" );
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document.write( "\"x%2Byi+=sqrt%28-5%2B12i%29\"\r\n" );
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document.write( "Squaring both sides:\r\n" );
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document.write( "\"%28x%2Byi%29%5E2+=+-5+%2B+12i\"\r\n" );
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document.write( "\"x%5E2%2B2xyi%2By%5E2i%5E2+=+-5+%2B+12i\"\r\n" );
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document.write( "\"x%5E2%2B2xyi%2By%5E2%28-1%29+=+-5+%2B+12i\"\r\n" );
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document.write( "\"x%5E2%2B2xyi-y%5E2+=+-5+%2B+12i\"\r\n" );
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document.write( "Set real parts equal:\r\n" );
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document.write( "\"x%5E2-y%5E2+=+-5\"\r\n" );
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document.write( "Set imaginary parts equal:\r\n" );
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document.write( "\"2xy+=+12\"\r\n" );
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document.write( "\"xy+=+6\"\r\n" );
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document.write( "Solve system:\r\n" );
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document.write( "\"system%28x%5E2-y%5E2+=+-5%2Cxy+=+6%29\"\r\n" );
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document.write( "Solve the second for a letter , substitute it in the first,\r\n" );
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document.write( "and get two solutions (x,y) = (2,3) and (x,y) = (-2,-3).\r\n" );
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document.write( "So \"+sqrt%28-5%2B12i%29=x+%2B+yi\" which can either be 2+3i or -2-3i.\r\n" );
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document.write( "Thus \"%22%22+%2B-+sqrt%28-5%2B12i%29=%22%22+%2B-+%282%2B3i%29\"\r\n" );
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document.write( "So the solution to\r\n" );
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document.write( "\"z%5E2%2Bzi%2B%281-3i%29=0\"\r\n" );
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document.write( "which was:\r\n" );
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document.write( "\"z+=+%28-i+%2B-+sqrt%28-5%2B12i%29+%29%2F2+\"\r\n" );
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document.write( "becomes\r\n" );
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document.write( "\"z+=+%28-i+%2B-+%282%2B3i%29+%29%2F2+\"\r\n" );
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document.write( "Using the +, that simplifies to 1+i\r\n" );
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document.write( "Using the -, that simplifies to -1-2i\r\n" );
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document.write( "So those are the two solutions to the quadratic.\r\n" );
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document.write( "Edwin
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