document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085367>Question 1085367</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all complex numbers z such that<BR>\n" );
document.write( "|z|^2-2\bar{z}+iz=2i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #699421 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=699421\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">I'm assuming the initial equation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE |z|^2-2\bar{z}+iz=2i\">\r<BR>\n" );
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document.write( "If so, then let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+a%2Bbi ALIGN=MIDDLE  ALT=\"z+=+a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a,b \in \mathbb{R}\"> (a,b are real numbers)\r<BR>\n" );
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document.write( "The complex conjugate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \bar{z} = a-bi\">\r<BR>\n" );
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document.write( "Using the pythagorean theorem, and a visual representation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z ALIGN=MIDDLE  ALT=\"z\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large |z|^2 = a^2+b^2\">\r<BR>\n" );
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document.write( "Using those equations, we can make a bit of substitutions and rearrangements to get the following\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE |z|^2-2\bar{z}+iz=2i\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2+b^2-2(a-bi)+i(a+bi)=2i\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2+b^2-2a+2bi+ai+bi^2=2i\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2+b^2-2a+2bi+ai+b(-1)=2i\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a^2+b^2-2a+2bi+ai-b=2i\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a^2+b^2-2a-b)+(2bi+ai)=2i\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (a^2+b^2-2a-b)+(2b+a)*i=0+2*i\">\r<BR>\n" );
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document.write( "From that last equation, in the section above, we can equate the real and imaginary parts to form these two equations\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2-2a-b+=+0 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2-2a-b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2Ba+=+2 ALIGN=MIDDLE  ALT=\"2b%2Ba+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2Ba+=+2 ALIGN=MIDDLE  ALT=\"2b%2Ba+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for 'a'\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2Ba+=+2 ALIGN=MIDDLE  ALT=\"2b%2Ba+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2Ba-2b+=+2-2b ALIGN=MIDDLE  ALT=\"2b%2Ba-2b+=+2-2b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2-2b ALIGN=MIDDLE  ALT=\"a+=+2-2b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now plug this into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2-2a-b+=+0 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2-2a-b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and solve for b\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2-2a-b+=+0 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2-2a-b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-2b%29%5E2%2Bb%5E2-2%282-2b%29-b+=+0 ALIGN=MIDDLE  ALT=\"%282-2b%29%5E2%2Bb%5E2-2%282-2b%29-b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-8b%2B4b%5E2%2Bb%5E2-4%2B4b-b+=+0 ALIGN=MIDDLE  ALT=\"4-8b%2B4b%5E2%2Bb%5E2-4%2B4b-b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284b%5E2%2Bb%5E2%29%2B%28-8b%2B4b-b%29%2B%284-4%29+=+0 ALIGN=MIDDLE  ALT=\"%284b%5E2%2Bb%5E2%29%2B%28-8b%2B4b-b%29%2B%284-4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5b%5E2+-+5b+=+0 ALIGN=MIDDLE  ALT=\"5b%5E2+-+5b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5b%28b+-+1%29+=+0 ALIGN=MIDDLE  ALT=\"5b%28b+-+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5b=0 ALIGN=MIDDLE  ALT=\"5b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+-+1+=+0 ALIGN=MIDDLE  ALT=\"b+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=0 ALIGN=MIDDLE  ALT=\"b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1 ALIGN=MIDDLE  ALT=\"b+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=0 ALIGN=MIDDLE  ALT=\"b=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2-2b ALIGN=MIDDLE  ALT=\"a+=+2-2b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2-2%2A0 ALIGN=MIDDLE  ALT=\"a+=+2-2%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2 ALIGN=MIDDLE  ALT=\"a+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Therefore one solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+a%2Bbi+=+2%2B0i+=+2 ALIGN=MIDDLE  ALT=\"z+=+a%2Bbi+=+2%2B0i+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is purely a real number.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1 ALIGN=MIDDLE  ALT=\"b+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2-2b ALIGN=MIDDLE  ALT=\"a+=+2-2b\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+2-2%2A1 ALIGN=MIDDLE  ALT=\"a+=+2-2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+0 ALIGN=MIDDLE  ALT=\"a+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The other solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+a%2Bbi+=+0%2B1i+=+i ALIGN=MIDDLE  ALT=\"z+=+a%2Bbi+=+0%2B1i+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is a <a  rel=nofollow HREF=\"http://mathworld.wolfram.com/PurelyImaginaryNumber.html\">purely imaginary number</a>\r<BR>\n" );
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document.write( "Summary:\r<BR>\n" );
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document.write( "The two solutions are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2%2B0i ALIGN=MIDDLE  ALT=\"z+=+2%2B0i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+2 ALIGN=MIDDLE  ALT=\"z+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+0%2B1i ALIGN=MIDDLE  ALT=\"z+=+0%2B1i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+i ALIGN=MIDDLE  ALT=\"z+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" >)</font> </SPAN>\n" );
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