document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=526130>Question 526130</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if x = y^2 and y = x^2, what is the least possible real value of x^2 + x + 1,<BR>\n" );
document.write( "where x and y are complex numbers ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #348500 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=348500\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = y^2 \Rightarrow x = (x^2)^2 \Rightarrow x = x^4 \Rightarrow x^3 = 1\"> (if x is not equal to 0). Hence the three possible values of x (roots of unity) are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = e^{0i} = 1\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = e^{\frac{2\pi}{3}i}\">, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x = e^{\frac{4\pi}{3}i}\">.\r<BR>\n" );
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document.write( "Hence, replacing each of these values into the expression x^2 + x + 1 and simplifying yields 3, 0, and 0 respectively, so the minimum value is 0. </SPAN>\n" );
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