document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=330684>Question 330684</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Look at the graph above and comment on the sign of the discriminant. Form the quadratic equation based on the information provided and find its solutions.<BR>\n" );
document.write( "    * This graph is a Parabola type. You can tell by its horse shoe shape and in a upright position<BR>\n" );
document.write( "    * The x2 coefficient term is positive, because of the upward opening. This tells me the formula ax^2 + bx + c = >0 is used. In addition, two real roots exist for a positive.<BR>\n" );
document.write( "    * Because the graph crosses the x axis, one or more real roots exist.<BR>\n" );
document.write( "    * A visible symmetrical (lowest point) exist and is called the vertex.  This position is on coordinates (-0.16,-2). Additionally, the vertex is located between a mirror image of the left and right lines. \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #237072 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=237072\">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( "Since there are two <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> intercepts, there are two distinct real number roots, hence the discriminant is positive.\r<BR>\n" );
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document.write( "However, there is insufficient information to determine the specific quadratic function or its zeros.\r<BR>\n" );
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document.write( "Using the coordinates of the vertex, you can determine two vital pieces of information. First since the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex of the general quadratic function is given by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{-b}{2a}\">, you can use the fact that the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0.16\"> to write a linear equation relating <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0.32a\ -\ b\ =\ 0\">\r<BR>\n" );
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document.write( "and you can also write another linear equation relating <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> by using the vertex coordinates since we know that any point on the function <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large f(x)\"> can be described as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x,\,f(x)\right)\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ a(-0.16)^2\ +\ b(-0.16) +\ c\ =\ -2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0.256a\ -\ 0.16b\ +\ c\ =\ -2\">\r<BR>\n" );
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document.write( "and by substitution:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -0.256a\ +\ c\ =\ -2\">\r<BR>\n" );
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document.write( "But that is as far as we can go without additional information.  The only thing that can be said for certain about the zeros of the function is that there exists a real number <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \alpha\"> such that the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts of the graph of your function are at:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(-0.16\,+\,\alpha,\,0\right)\">\r<BR>\n" );
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document.write( "and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(-0.16\,-\,\alpha,\,0\right)\">\r<BR>\n" );
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document.write( "The coordinates of any other point on the graph of your function would be sufficient to uniquely determine the coefficients.\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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