document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=444589>Question 444589</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #306433 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=306433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "Let's assume that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{\alpha}{\rho}\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{\beta}{\sigma}\"> are the roots of some quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2\ +\ bx\ +\ c\ =\ 0\">\r<BR>\n" );
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document.write( "Then with a little algebra we can see that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \rho{x}\ -\ \alpha\ =\ 0\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \sigma{x}\ -\ \beta\ =\ 0\">.\r<BR>\n" );
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document.write( "Reversing the Zero Product Rule, we can write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\rho{x}\ -\ \alpha\right)\left(\sigma{x}\ -\ \beta\right)\ =\ 0\">\r<BR>\n" );
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document.write( "Apply FOIL\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho\sigma{x^2}\ +\ \left(\beta\rho\ +\ \alpha\sigma\right)x\ +\ \alpha\beta\ =\ 0\">\r<BR>\n" );
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document.write( "Then let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ \rho\sigma\ \ \">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ \beta\rho\ +\ \alpha\sigma\ \ \">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\ =\ \alpha\beta\"> and substitute:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ +\ c\ =\ 0\">\r<BR>\n" );
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document.write( "The only fly in the ointment is that you can take any real number, say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> and apply it as a factor:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k\left(\rho{x}\ -\ \alpha\right)\left(\sigma{x}\ -\ \beta\right)\ =\ 0\">\r<BR>\n" );
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document.write( "You still have the same roots to the equation, but when you multiply it out you get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ kax^2\ +\ kbx\ +\ kc\ =\ 0\">\r<BR>\n" );
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document.write( "Which means that there are an infinite number of quadratic equations with a given pair of roots (or given root with a muliplicity of 2) but they only differ by a factor common to all three terms in the trinomial representation of the quadratic.\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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