document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=577452>Question 577452</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how many quadratic equations are there whose roots when squared gives the same quadratic equation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #370292 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=370292\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Suppose the roots are p and q. Therefore we want\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = (x-p)(x-q) = (x-p^2)(x-q^2)\">.\r<BR>\n" );
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document.write( "Hence either p = p^2 and q = q^2, or p = q^2 and q = p^2. The first case implies that p and q must be either 0 or 1, so we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = x^2\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = x(x-1)\"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = (x-1)^2\">\r<BR>\n" );
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document.write( "The second case implies p = p^4, or p^3 = 1. The roots for p are 1, -1/2 + i*sqrt(3) and -1/2 - i*sqrt(3)/2 (same for q).\r<BR>\n" );
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document.write( "We have taken care of the p = 1 case, so suppose that p is -1/2 + i*sqrt(3)/2. Then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q = p^2 = (-\frac{1}{2} + \frac{i \sqrt{3}}{2})^2 = -\frac{1}{2} - \frac{i \sqrt{3}}{2}\"> (easy way to square it is note that p = e^(2i*pi/3))\r<BR>\n" );
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document.write( "Therefore the polynomial f(x) is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f(x) = (x - (-\frac{1}{2} + \frac{i \sqrt{3}}{2}))(x - (-\frac{1}{2} - \frac{i \sqrt{3}}{2})) = x^2 - x - 1\">\r<BR>\n" );
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document.write( "So there are four quadratic equations that satisfy (ignoring arbitrary constants we can multiply with). </SPAN>\n" );
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