document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=372140>Question 372140</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If I have a solution how can I create a quadric equation  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #265084 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=265084\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You can't, in general.  That is because you need two solutions to create a quadratic.  However, in some cases, the character of the solution that you do have allows you to deduce the solution that you don't have.\r<BR>\n" );
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document.write( "For example, if you have a solution that looks like\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1\ +\ \sqrt{3}}{2}\">\r<BR>\n" );
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document.write( "Then, because of symmetry, the other solution is guaranteed to be:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{1\ -\ \sqrt{3}}{2}\">.\r<BR>\n" );
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document.write( "Likewise, if you have a single complex number solution, then the other solution is the conjugate of that complex number.\r<BR>\n" );
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document.write( "On the other hand, if you only have one rational number solution, you are at a dead end, unless you know that your single rational solution has a multiplicity of two.\r<BR>\n" );
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document.write( "Given that you do have two solutions, let's call them <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \beta\">, all you have to do is calculate the product:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ \alpha)(x\ -\ \beta)\">\r<BR>\n" );
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document.write( "And set that product equal to zero.\r<BR>\n" );
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document.write( "In the case where you have one solution with a multiplicity of two, meaning you have two identical solutions, you can calculate:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ \alpha)^2\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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