document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=42640>Question 42640</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The demand and supply equations for a certain item are given by\r<BR>\n" );
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document.write( "D = -5p + 40<BR>\n" );
document.write( "S =  -p^2 + 30p - 8\r<BR>\n" );
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document.write( "Find the equilibrium price. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #27750 by </B><A HREF=/tutors/aboutme.mpl?userid=Nate><B>Nate(3500)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Nate><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=27750\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > as total payment:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+-5p+%2B+40 ALIGN=MIDDLE  ALT=\"D+=+-5p+%2B+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "also known as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-5p+%2B+40 ALIGN=MIDDLE  ALT=\"y+=+-5p+%2B+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=++-p%5E2+%2B+30p+-+8 ALIGN=MIDDLE  ALT=\"S+=++-p%5E2+%2B+30p+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "also known as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-p%5E2+%2B+30p+-+8 ALIGN=MIDDLE  ALT=\"y+=+-p%5E2+%2B+30p+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Combine:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-p%5E2+%2B+30p+-+8 ALIGN=MIDDLE  ALT=\"y+=+-p%5E2+%2B+30p+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > while you know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-5p+%2B+40 ALIGN=MIDDLE  ALT=\"y+=+-5p+%2B+40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5p+%2B+40+=+-p%5E2+%2B+30p+-+8 ALIGN=MIDDLE  ALT=\"-5p+%2B+40+=+-p%5E2+%2B+30p+-+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-p%5E2+%2B+35p+-+48 ALIGN=MIDDLE  ALT=\"0+=+-p%5E2+%2B+35p+-+48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<TABLE  BORDER=1>\n" );
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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=ap%5E2%2Bbp%2Bc=0 ALIGN=MIDDLE  ALT=\"ap%5E2%2Bbp%2Bc=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (in our case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1p%5E2%2B35p%2B-48+=+0 ALIGN=MIDDLE  ALT=\"-1p%5E2%2B35p%2B-48+=+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=p%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"p%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=%2835%29%5E2-4%2A-1%2A-48=1033 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2835%29%5E2-4%2A-1%2A-48=1033\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1033 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+=+%28-35%2B-sqrt%28+1033+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-35%2B-sqrt%28+1033+%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=p%5B1%5D+=+%28-%2835%29%2Bsqrt%28+1033+%29%29%2F2%5C-1+=+1.4298413200118 ALIGN=MIDDLE  ALT=\"p%5B1%5D+=+%28-%2835%29%2Bsqrt%28+1033+%29%29%2F2%5C-1+=+1.4298413200118\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5B2%5D+=+%28-%2835%29-sqrt%28+1033+%29%29%2F2%5C-1+=+33.5701586799882 ALIGN=MIDDLE  ALT=\"p%5B2%5D+=+%28-%2835%29-sqrt%28+1033+%29%29%2F2%5C-1+=+33.5701586799882\"   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=-1p%5E2%2B35p%2B-48 ALIGN=MIDDLE  ALT=\"-1p%5E2%2B35p%2B-48\"   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=-1p%5E2%2B35p%2B-48+=+-1%28p-1.4298413200118%29%2A%28p-33.5701586799882%29 ALIGN=MIDDLE  ALT=\"-1p%5E2%2B35p%2B-48+=+-1%28p-1.4298413200118%29%2A%28p-33.5701586799882%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 1.4298413200118, 33.5701586799882.\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+-1%2Ax%5E2%2B35%2Ax%2B-48+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B35%2Ax%2B-48+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-5%2C+35%2C+-150%2C+100%2C+-5x+%2B+40%2C+-x%5E2+%2B+30x+-+8%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-5%2C+35%2C+-150%2C+100%2C+-5x+%2B+40%2C+-x%5E2+%2B+30x+-+8%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );