document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1066706>Question 1066706</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the equation (z-1)^3 + ((i-1)/(i+1))^3 =0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #681964 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=681964\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (z - 1)^3 + ((i - 1) / (i + 1))^3 = 0<BR>\n" );
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document.write( "Note that ((i - 1) / (i + 1))^3 = -i, since (i - 1) / (i + 1) = i then i^3 = i^2 * i = -1 * i = -i<BR>\n" );
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document.write( "(z - 1)^3 -i = 0<BR>\n" );
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document.write( "(z - 1)^3  = i<BR>\n" );
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document.write( "Note i = (-i)^3 and let x = (z - 1), then<BR>\n" );
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document.write( "x^3 - (-1)^3 = 0<BR>\n" );
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document.write( "we know that a^3 - b^3 = (a - b) * (a^2 +ab + b^2), then<BR>\n" );
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document.write( "{x&#8722;(&#8722;i)} * {x^2+x(&#8722;i)+(&#8722;i)^2} = {x&#8722;(&#8722;i)} * (x^2&#8722;ix&#8722;1) = 0<BR>\n" );
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document.write( "If x&#8722;(&#8722;i) = 0, x = &#8722;i, now x = z - 1 then<BR>\n" );
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document.write( "z - 1 = -i<BR>\n" );
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document.write( "z = 1 - i<BR>\n" );
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document.write( "Else x^2 &#8722;ix &#8722;1 = 0<BR>\n" );
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document.write( "use the quadratic formula<BR>\n" );
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document.write( "x = (i + or - square root(i^2 -(4)(-1))) / 2 = (i + or - square root(3)) / 2)<BR>\n" );
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document.write( "z - 1 = (i + or - square root(3)) / 2)<BR>\n" );
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document.write( "z = 1 + (i + or - square root(3)) / 2<BR>\n" );
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document.write( "Note there are 3 solutions for z<BR>\n" );
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document.write( "z = 1 - i<BR>\n" );
document.write( "z = 1 + (i + square root(3)) / 2<BR>\n" );
document.write( "z = 1 + (i - square root(3)) / 2<BR>\n" );
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document.write( " </SPAN>\n" );
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