document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=340006>Question 340006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do you know if a quadratic equation will have one, two, or no solutions? 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? Explain. Provide your classmate’s with one or two solutions with which they must create a 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 #243678 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=243678\">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( "The first part of your question is easy.  There are ALWAYS two solutions, if you count the multiplicity.  The Fundamental Theorem of Algebra says so.  However, evaluate the discriminant to determine the nature of the roots.\r<BR>\n" );
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document.write( "For any quadratic polynomial equation of the form:\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( "Find the Discriminant, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta\ =\ b^2\ -\ 4ac\"> and evaluate the nature of the roots as follows:\r<BR>\n" );
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document.write( "No calculation quick look:  If the signs on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> are opposite, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta > 0\"> guaranteed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta > 0 \ \ \Rightarrow\ \\"> Two real and unequal roots. If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta\"> is a perfect square, the quadratic factors over <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{Q}\">.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta = 0 \ \ \Rightarrow\ \\"> One real root with a multiplicity of two.  That is to say that the trinomial is a perfect square and has two identical factors.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta < 0 \ \ \Rightarrow\ \\"> A conjugate pair of complex roots of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a \pm bi\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE i\"> is the imaginary number defined by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE i^2 = -1\">\r<BR>\n" );
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document.write( "+----------+----------+----------+----------+----------+----------+----------+\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \alpha\">\r<BR>\n" );
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document.write( "is a solution of the quadratic equation:\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( "if and only if\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ \alpha\">\r<BR>\n" );
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document.write( "is a factor of the quadratic polynomial:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ +\ c\">\r<BR>\n" );
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document.write( "So, just take your given solution, whatever it is, and form a binomial factor by subtracting from x.  Do this for each of your solutions.  If you are told there is only one solution, say, out loud, \"Ok, there is a pair of identical solutions\" and create two identical factors.  If you are given one irrational solution, remember that irrational solutions always come in conjugate pairs.  If you are given <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ +\ \sigma\sqrt{\rho}\"> as a solution, then you know that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \varphi\ -\ \sigma\sqrt{\rho}\"> must also be a solution. The conjugate trick works for complex roots (the ones that involve <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large i\"> that you get when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta < 0\">) also.\r<BR>\n" );
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document.write( "Once you have your two factors, multiply them together using our old friend FOIL, and set the result equal to zero.  That will give you one possible quadratic equation with the given roots.\r<BR>\n" );
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document.write( "It is most assuredly possible.  In fact, for any pair of solutions, real or otherwise, there is a set of quadratic equations with infinite elements.\r<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \{\alpha,\beta\}\"> is the solution set of the quadratic equation\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( "Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \{\alpha,\beta\}\"> is the solution set of every quadratic equation of the form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ akx^2\ +\ bkx\ +\ ck\ = 0\">\r<BR>\n" );
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document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE k\ \in\ \mathbb{R}\">\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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