Algebra.Com's Answer #833684 by ikleyn(52797)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "2) find the set solution of:
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document.write( " iz² - i³|2z²| = 3i×(conj(z))², such z belong complex numbers
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document.write( " In this post, I will solve your second problem, ONLY.\r
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document.write( "Our starting equation is\r\n" );
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document.write( "  -  =  . (1)\r\n" );
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document.write( "Since  = -1, this equation is the same as\r\n" );
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document.write( " +  = .\r\n" );
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document.write( "We can write it in this equivalent form\r\n" );
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document.write( " = - .\r\n" );
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document.write( "We can cancel common factor \" i \" in both sides\r\n" );
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document.write( "  =  -  . (2)\r\n" );
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document.write( "Left side is a real number --- hence, right side must be a real number, too.\r\n" );
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document.write( "Ok. Now let z = a+bi. Then from (2), right side of (2) is\r\n" );
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document.write( "  -  = 3*(a^2 - 2abi - b^2) - (a^2 +2abi - b^2) =\r\n" );
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document.write( " = (3a^2 - 3b^2- a^2 + b^2) + (-6abi - 2abi) = (2a^2 - 2b^2) - 8abi. (3)\r\n" );
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document.write( "Since the right side of (2) is a real number, it implies from (3) that ab = 0, i.e.\r\n" );
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document.write( " EITHER a= 0 OR b= 0 (or BOTH).\r\n" );
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document.write( "So, below we analyze two cases.\r\n" );
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document.write( " (a) b= 0. It means that \"z\" is, actually, a real number.\r\n" );
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document.write( " It is easy to check, that then equation (2) is valid for any value of real number z,\r\n" );
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document.write( " which means that any real number is the solution for the original equation.\r\n" );
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document.write( " (b) a= 0, b=/= 0. It means that \"z\" is, actually, a pure imaginary number.\r\n" );
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document.write( " Then in (2), the left side is positive real number, while the right side (3) \r\n" );
document.write( " is negative real number, which creates a CONTRADICTORY.\r\n" );
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document.write( " | So, the  to the problem is that |\r\n" );
document.write( " | the solution set for the given equation is the set of all real numbers. |\r\n" );
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document.write( "Solved, with full explanations.\r
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