document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=633133>Question 633133</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number of coffee shops in a certain country can be modeled by the quadratic function f(x)=111x^2+680x+4486 where f(x) is the number of the coffee shops and x is the number of years after 2000. Two part answer - part a) the number of coffee shops in 2004 was? and b) the year in which the number of coffee shops will be 20,000 is?<BR>\n" );
document.write( "I've tried trying applying the quadratic formula for the first part, but I don't understand the two parts they want answered. Thank you for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #398680 by </B><A HREF=/tutors/aboutme.mpl?userid=graphmatics><B>graphmatics(170)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=graphmatics><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=graphmatics><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=398680\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve part a) compute f(4)=111*4^2+680*4+4486=1776+2720+4486=8982. To solve part b) find x from f(x)=20000. 111*x^2+680*x+4486=20000 111*x^2+680*x-15514=0 <TABLE  BORDER=1>\n" );
document.write( "\n" );
document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=quadratic.solver>SOLVE quadratic equation with variable</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Quadratic equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%5E2%2Bbx%2Bc=0 ALIGN=MIDDLE  ALT=\"ax%5E2%2Bbx%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111x%5E2%2B680x%2B-15514+=+0 ALIGN=MIDDLE  ALT=\"111x%5E2%2B680x%2B-15514+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) has the following solutons:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For these solutions to exist, the <U>discriminant</U> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" > should not be a negative number.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, we need to compute the discriminant <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac ALIGN=MIDDLE  ALT=\"b%5E2-4ac\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-4ac=%28680%29%5E2-4%2A111%2A-15514=7350616 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28680%29%5E2-4%2A111%2A-15514=7350616\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=7350616 is greater than zero. That means that there are two solutions: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5B12%5D+=+%28-680%2B-sqrt%28+7350616+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-680%2B-sqrt%28+7350616+%29%29%2F2%5Ca\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D+=+%28-%28680%29%2Bsqrt%28+7350616+%29%29%2F2%5C111+=+9.14955832196591 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28680%29%2Bsqrt%28+7350616+%29%29%2F2%5C111+=+9.14955832196591\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D+=+%28-%28680%29-sqrt%28+7350616+%29%29%2F2%5C111+=+-15.275684448092 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28680%29-sqrt%28+7350616+%29%29%2F2%5C111+=+-15.275684448092\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <BR>\n" );
document.write( "    Quadratic expression <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111x%5E2%2B680x%2B-15514 ALIGN=MIDDLE  ALT=\"111x%5E2%2B680x%2B-15514\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be factored:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=111x%5E2%2B680x%2B-15514+=+111%28x-9.14955832196591%29%2A%28x--15.275684448092%29 ALIGN=MIDDLE  ALT=\"111x%5E2%2B680x%2B-15514+=+111%28x-9.14955832196591%29%2A%28x--15.275684448092%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 9.14955832196591, -15.275684448092.\n" );
document.write( "Here's your graph:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+111%2Ax%5E2%2B680%2Ax%2B-15514+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+111%2Ax%5E2%2B680%2Ax%2B-15514+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );