document.write( "Question 1181502: What are the roots of (z^2) - z(1 + i) + 5i = 0? \n" ); document.write( "
Algebra.Com's Answer #811409 by ikleyn(52781)\"\" \"About 
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\n" ); document.write( "What are the roots of (z^2) - z(1 + i) + 5i = 0?
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document.write( "What you see in front of you, is a quadratic equation over complex numbers with complex coefficients.\r\n" );
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document.write( "Exactly as usual quadratic equation over real numbers, you can solve it over complex numbers using the Quadratic Formula.\r\n" );
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document.write( "    \"z%5B1%2C2%5D\" = \"%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\" = \"%28%281%2Bi%29+%2B-+sqrt%28%281%2Bi%29%5E2+-+4%2A%285i%29%29%29%2F2\" = \"%28%281%2Bi%29+%2B-+sqrt%281%2B2i%2Bi%5E2+-20i%29%29%2F2\" = \r\n" );
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document.write( "  = \"%28%281%2Bi%29+%2B-+sqrt%28-18i%29%29%2F2\" = \"%28%281%2Bi%29+%2B-+3%2Asqrt%28-2i%29%29%2F2\".    (1)\r\n" );
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document.write( "Now notice that  -2i = 2*cis(270°);  therefore,  \"sqrt%28-2i%29\" = \"sqrt%282%29%2Acis%28135%5Eo%29\" = (-1+i).\r\n" );
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document.write( "Having it, we can continue the formula (1) this way\r\n" );
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document.write( "    \"z%5B1%2C2%5D\" = \"%28%281%2Bi%29+%2B-+3%2A%28-1%2Bi%29%29%2F2\".\r\n" );
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document.write( "Therefore,\r\n" );
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document.write( "    \"z%5B1%5D\" = \"%28%281%2Bi%29+%2B+3%2A%28-1%2Bi%29%29%2F2\" = \"%281%2Bi+-+3+%2B+3i%29%2F2\" = \"%28-2+%2B+4i%29%2F2\" = -1 + 2i;\r\n" );
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document.write( "    \"z%5B2%5D\" = \"%28%281%2Bi%29+-+3%2A%28-1%2Bi%29%29%2F2\" = \"%281%2Bi+%2B+3+-+3i%29%2F2\" = \"%284+-+2i%29%2F2\" = 2 - i.\r\n" );
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document.write( "ANSWER.  The roots are  -1 + 2i  and  2 - i.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Compare the volume of my calculations with that by @MathLover1.\r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of my lessons on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
\n" ); document.write( "\n" ); document.write( "    - Complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with real coefficients on complex domain\r
\n" ); document.write( "\n" ); document.write( "    - How to take a square root of a complex number\r
\n" ); document.write( "\n" ); document.write( "    - Solution of the quadratic equation with complex coefficients on complex domain\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problem on taking square root of complex number\r
\n" ); document.write( "\n" ); document.write( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on de'Moivre formula\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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