document.write( "Question 1076735: What is the number of distinct solution of the equation $\"z%5E2%2Babs%28z%29\"$ = $\"0\"$ ? \n" ); document.write( "

Algebra.Com's Answer #691355 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "What is the number of distinct solution of the equation $\"z%5E2%2Babs%28z%29\"$ = $\"0\"$ ?
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document.write( "Let us write the number z in Trigonometry form: z = .\r\n" );
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document.write( "Then  = , and your equation is\r\n" );
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document.write( " = ,   or, which is the same,\r\n" );
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document.write( " = 0.\r\n" );
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document.write( "The last equation deploys in two independent equations\r\n" );
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document.write( "1)  r = 0,   which means simply  z = 0.\r\n" );
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document.write( "2)  (r*(cos(2a) + i*sin(2a)) + 1 = 0,  which is the same as\r\n" );
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document.write( "    r*(cos(2a) + i*sin(2a) = -1.\r\n" );
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document.write( "    The last equation implies \r\n" );
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document.write( "    r = 1,  cos(2a) = -1 and sin(2a) = 0,   which, in turn, implies\r\n" );
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document.write( "    r = 1  and  {2a = pi   OR  2a =  }.\r\n" );
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document.write( "    \r\n" );
document.write( "    It means  that the solutions are  \r\n" );
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document.write( "           a) r = 1,  a = ,     OR/AND\r\n" );
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document.write( "           b) r = 1,  a = .\r\n" );
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document.write( "    In the rectangular form these solutions are  z = i   OR/AND  z = -i.\r\n" );
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document.write( "Answer.  The original equation has three solutions: z = 0;  z = i,  and  z = -i.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is a bunch of 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
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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( "    - Miscellaneous problems on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problem on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - A curious example of an equation in complex numbers which HAS NO a solution\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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