document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1188454>Question 1188454</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A quadratic function has roots 3 and - 5, which passes through the point (1, 5). Under what conditions will it intersect with a line of slope -4 once? Twice? Never? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #819550 by </B><A HREF=/tutors/aboutme.mpl?userid=Solver92311><B>Solver92311(821)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Solver92311><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=Solver92311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=819550\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Sunday, April 2, 2017<BR>\n" );
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document.write( "Quadratic with roots 3 and -5 that passes through (1,5).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ a(x\,-\,3)(x\,+\,5)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(1)\ =\ a(-2)(6)\ =\ 5\">\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ -\frac{5}{12}\"> and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \rho(x)\ =\ -\frac{5}{12}x^2\ -\ \frac{5}{6}x\ +\ \frac{25}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{d\rho}{dx}\ =\ -\frac{5}{6}x\ -\ \frac{5}{6}\">\r<BR>\n" );
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document.write( "Set the first derivative equal to -4 and solve:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -\frac{5}{6}x\ -\ \frac{5}{6}\ =\ -4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ +\ 1\ =\ -4\(-\frac{6}{5}\)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ \frac{19}{5}\">\r<BR>\n" );
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document.write( "All lines with slope -4 have an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ -4(x\,-\,x_1)\ +\ \rho(x_1)\">.  Since a line with a slope of -4 is tangent to the graph of the quadratic function at the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \(\frac{19}{5},\rho\(\frac{19}{5}\)\)\">, or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \(\frac{19}{5},-\frac{44}{15}\)\">, the line <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ -4\(x\,-\,\frac{19}{5}\)\ -\ \frac{44}{15}\"> intersects the function graph exactly once.  Any line of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ =\ -4(x\,-\,x_1)\ +\ \rho(x_1)\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_1\ >\ \frac{19}{5}\"> will not intersect the function graph anywhere. And where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_1\ <\ \frac{19}{5}\">, the line will intersect the function graph in two places.\r<BR>\n" );
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document.write( "																<BR>\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( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> <BR>\n" );
document.write( "I > Ø<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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