document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=204107>Question 204107</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could someone please help me with this proble?\r<BR>\n" );
document.write( "\n" );
document.write( "Determine the number of solutions and classify the type of solutions for the following equation and justify the answer.<BR>\n" );
document.write( " <BR>\n" );
document.write( "x^2 + 3x - 15 = 0 <BR>\n" );
document.write( "my answer is that there are two solutions x = 2.65, -5.65 but I'm not sure how to classify the type of solution and justify it. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #154069 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=154069\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The Fundamental Theorem of Algebra says that any polynomial equation of the form:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \alpha_0x^n\,+\,\alpha_1x^{n-1}\,+\,\cdots\,+\,\alpha_{n-1}x^1\,+\,\alpha_nx^0\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Has <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> roots.  So your equation must have two roots.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The Discriminant, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta\">, is the portion of the quadratic equation under the radical in the quadratic equation, namely:\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Delta = b^2\,-\,4ac\">, where the given equation is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large ax^2\, +\, bx\, +\ c\ =\ 0\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Classify the roots as follows.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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 not a perfect square, then the two roots are a conjugate pair of irrational numbers of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha \pm \beta\gamma\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha,\,\beta\,\in\,Q\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \gamma\,\in\,\R,\ \notin\,Q\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "<BR>\n" );
document.write( "\n" );
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 \alpha \pm \beta i\"> 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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "By the way, your claim that the roots are <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 2.65\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ -5.65\"> is not precisely correct.  If you want to express the roots as decimals, the best you can do is express them as an approximation.  So you should have said <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ \approx\ 2.65\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ \approx\ -5.65\">.  The only way to express the roots exactly is to leave the answers in radical form, namely <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ \frac{9\,\pm\,\sqrt{69}}{2}\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );