document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=446552>Question 446552</A>: <I><SPAN STYLE=\"word-break: break-all;\"> One machine works 8 hours faster then a second machine. If the two machines take twelve hours to finish the job working together, how long will it take them individually. <BR>\n" );
document.write( "I just need to figure out how to get to aquadratic equasion.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #307535 by </B><A HREF=/tutors/aboutme.mpl?userid=nerdybill><B>nerdybill(7384)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=nerdybill><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=nerdybill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=307535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One machine works 8 hours faster then a second machine. If the two machines take twelve hours to finish the job working together, how long will it take them individually. <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let x = time (hrs) it takes for slower machine<BR>\n" );
document.write( "then<BR>\n" );
document.write( "x-8 = time of faster machine<BR>\n" );
document.write( ".<BR>\n" );
document.write( "12(1/x + 1/(x-8)) = 1<BR>\n" );
document.write( "multiplying both sides by x(x-8):<BR>\n" );
document.write( "12(x-8 + x) = x(x-8)<BR>\n" );
document.write( "12(2x-8) = x(x-8)<BR>\n" );
document.write( "24x-96 = x^2-8x<BR>\n" );
document.write( "-96 = x^2-32x<BR>\n" );
document.write( "0 = x^2-32x+96<BR>\n" );
document.write( "applying the quadratic equation we get:<BR>\n" );
document.write( "x = {28.65, 3.35}<BR>\n" );
document.write( "we can throw out the 3.35 leaving:<BR>\n" );
document.write( "x = 28.65 hrs (slower machine)<BR>\n" );
document.write( ".<BR>\n" );
document.write( "faster machine:<BR>\n" );
document.write( "x-8 = 28.65-8 = 20.65 hrs<BR>\n" );
document.write( ".<BR>\n" );
document.write( "detail of quadratic follows:<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=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=1x%5E2%2B-32x%2B96+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-32x%2B96+=+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=%28-32%29%5E2-4%2A1%2A96=640 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-32%29%5E2-4%2A1%2A96=640\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=640 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--32%2B-sqrt%28+640+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--32%2B-sqrt%28+640+%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-%28-32%29%2Bsqrt%28+640+%29%29%2F2%5C1+=+28.6491106406735 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-32%29%2Bsqrt%28+640+%29%29%2F2%5C1+=+28.6491106406735\"   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-%28-32%29-sqrt%28+640+%29%29%2F2%5C1+=+3.35088935932648 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-32%29-sqrt%28+640+%29%29%2F2%5C1+=+3.35088935932648\"   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=1x%5E2%2B-32x%2B96 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-32x%2B96\"   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=1x%5E2%2B-32x%2B96+=+1%28x-28.6491106406735%29%2A%28x-3.35088935932648%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-32x%2B96+=+1%28x-28.6491106406735%29%2A%28x-3.35088935932648%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 28.6491106406735, 3.35088935932648.\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%2B-32%2Ax%2B96+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-32%2Ax%2B96+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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