document.write( "Question 1073485: z-|z|=8+4i
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Algebra.Com's Answer #688313 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "Let z = a + bi be our complex number under the question.\r\n" );
document.write( "We need to find its real and imaginary parts \"a\" and \"b\".\r\n" );
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document.write( "As you, probably, know  |z| = .\r\n" );
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document.write( "So, the given equation becomes\r\n" );
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document.write( "a + bi -  = 8 + 4i.\r\n" );
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document.write( "It actually deploys in two independent equations for the real part and the imaginary part separately:\r\n" );
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document.write( "a -  = 8     (1)     and\r\n" );
document.write( "b            = 4.    (2)\r\n" );
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document.write( "Equation (1) is equivalent to\r\n" );
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document.write( "a - 8 = ,\r\n" );
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document.write( " = ,\r\n" );
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document.write( " = ,    (I substituted b= 4)\r\n" );
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document.write( "-16a + 64 = 16  --->  -16a = 16-64  --->  -16a = -48  ---> a = 3.\r\n" );
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document.write( "So, the logic leads us to the solution  a = 3, b = 4, i.e. z = 3 + 4i.\r\n" );
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document.write( "Now CHECK this solution: 3 + 4i -  = 3 + 4i - 5 = -2 + 4i.\r\n" );
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document.write( "It doesn't check !!!\r\n" );
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document.write( "The conclusion is: The given equation HAS NO solutions.\r\n" );
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\n" ); document.write( "When I got this result, I was very surprised: it is for the first time in my life I see the equation for complex numbers,
\n" ); document.write( "which has no solution.\r
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\n" ); document.write( "\n" ); document.write( "But reviewing the problem and the solution, I got the understanding WHY it happened in this case.\r
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\n" ); document.write( "\n" ); document.write( "The equation of the form z - |z| = u + iw CAN NOT have a solution if the real part \"u\" of the complex number
\n" ); document.write( "in the right hand side is positive.\r
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\n" ); document.write( "\n" ); document.write( "Actually, it is OBVIOUS !\r
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\n" ); document.write( "\n" ); document.write( "Now I am calm. Because I know the reason.
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\n" ); document.write( "\n" ); document.write( "See my lessons on complex numbers in this site \r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetic 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
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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
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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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