document.write( "Question 888022: If z1+z2+z3=0 and z1^2+z2^2+z3^2=0, prove that |z1|=|z2|=|z3|
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Algebra.Com's Answer #537043 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"z%5B1%5D%2Bz%5B2%5D%2Bz%5B3%5D\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B3%5D\"\"%22%22=%22%22\"\"-z%5B2%5D-z%5B1%5D\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2Bz%5B3%5D%5E2\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2B%28-z%5B2%5D-z%5B1%5D%29%5E2\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2B%28z%5B2%5D%5E2%2B2z%5B1%5Dz%5B2%5D%2Bz%5B1%5D%5E2%29\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2Bz%5B2%5D%5E2%2B2z%5B1%5Dz%5B2%5D%2Bz%5B1%5D%5E2\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"2z%5B1%5D%5E2%2B2z%5B2%5D%5E2%2B2z%5B1%5Dz%5B2%5D\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B1%5D%5E2%2Bz%5B2%5D%5E2%2Bz%5B1%5Dz%5B2%5D\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D%5E2%2Bz%5B1%5Dz%5B2%5D%2Bz%5B1%5D%5E2\"\"%22%22=%22%22\"\"%220%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D\"\"%22%22=%22%22\"\"%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D\"\"%22%22=%22%22\"\"%28-z%5B1%5D+%2B-+sqrt%28z%5B1%5D%5E2-4%2A1%2Az%5B1%5D%5E2+%29%29%2F%282%2A1%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D\"\"%22%22=%22%22\"\"%28-z%5B1%5D+%2B-+sqrt%28z%5B1%5D%5E2-4z%5B1%5D%5E2+%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D\"\"%22%22=%22%22\"\"%28-z%5B1%5D+%2B-+sqrt%28-3z%5B1%5D%5E2%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D\"\"%22%22=%22%22\"\"%28-z%5B1%5D+%2B-+z%5B1%5D%2Ai%2Asqrt%283%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"z%5B2%5D\"\"%22%22=%22%22\"\"z%5B1%5D%28-1+%2B-+i%2Asqrt%283%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+%2A+abs%28+%28-1+%2B-+i%2Asqrt%283%29%29%2F2%29+%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+%2A+abs%28+-1%2F2+%2B-+i%2Asqrt%283%29%2F2+%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+%2A+sqrt%28%28-1%2F2%29%5E2+%2B%28%22%22+%2B-+sqrt%283%29%2F2%29%5E2+%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+%2A+sqrt%281%2F4+%2B3%2F4+%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+%2A+sqrt%281+%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+%2A%281+%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B2%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+\"\r\n" );
document.write( "\r\n" );
document.write( "by symmetry, we may swap \"z%5B2%5D\" and \"z%5B3%5D\"\r\n" );
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document.write( "\"abs%28z%5B3%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B1%5D%29+\"\r\n" );
document.write( "\r\n" );
document.write( "\"abs%28z%5B1%5D%29\"\"%22%22=%22%22\"\"+abs%28z%5B2%5D%29+\"\"%22%22=%22%22\"\"+abs%28z%5B3%5D%29+\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
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