document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=223755>Question 223755</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A boat travels 9km upstream and 9km back.  The time round trip takes 10hrs.  The speed of the stream is 2km/hr.  What is the speed of the boat in still water? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #167367 by </B><A HREF=/tutors/aboutme.mpl?userid=MLipsky><B>MLipsky(9)</B></A><img src=\"/images/stars/stars-05.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MLipsky><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=167367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <hr /> For this problem, we need the distance formula:  rt=d.\r<BR>\n" );
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document.write( "The current is 2 km/h.  Even though we don't know the rate of the boat in still water, we know that going upstream, the rate can be expressed as:  (r-2).  Downstream, the rate is (r+2).\r<BR>\n" );
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document.write( "Now let's deal with \"time\": <BR>\n" );
document.write( "(t-upstream) + (t-downstream) = 10<BR>\n" );
document.write( "Rewrite: (t-downstream) = 10 - (t-upstream)<BR>\n" );
document.write( "now, we can use that second equation to replace (t-downstream) in a minute.\r<BR>\n" );
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document.write( "Getting back to the original formula: rt=d, we get 2 equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-2%29%28t%29=9 ALIGN=MIDDLE  ALT=\"%28r-2%29%28t%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B2%29%2810-t%29=9 ALIGN=MIDDLE  ALT=\"%28r%2B2%29%2810-t%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There is our system of equations.  Let's solve it with substitution.  We can solve the first equation in terms of \"t\":<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=9%2F%28r-2%29 ALIGN=MIDDLE  ALT=\"t=9%2F%28r-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (divided both sides by (r-2) )\r<BR>\n" );
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document.write( "Substitute that for \"t\" in the second equation.  (This gets long-haired.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B2%29%2810-9%2F%28r-2%29%29=9 ALIGN=MIDDLE  ALT=\"%28r%2B2%29%2810-9%2F%28r-2%29%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r-9r%2F%28r-2%29%2B20-18%2F%28r-2%29=9 ALIGN=MIDDLE  ALT=\"10r-9r%2F%28r-2%29%2B20-18%2F%28r-2%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (FOIL)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r%2B%28-9r-18%29%2F%28r-2%29%2B20=9 ALIGN=MIDDLE  ALT=\"10r%2B%28-9r-18%29%2F%28r-2%29%2B20=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (added fractions)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r%2B%28-9r-18%29%2F%28r-2%29=-11 ALIGN=MIDDLE  ALT=\"10r%2B%28-9r-18%29%2F%28r-2%29=-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > (subtracted 20 from both sides)<BR>\n" );
document.write( "Let's finally get rid of that complex fractions.  Multiply every term by (r-2).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r%28r-2%29%2B%28-9r-18%29=-11%28r-2%29 ALIGN=MIDDLE  ALT=\"10r%28r-2%29%2B%28-9r-18%29=-11%28r-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r%5E2-20r-9r-18=-11r%2B22 ALIGN=MIDDLE  ALT=\"10r%5E2-20r-9r-18=-11r%2B22\"   DISABLED_style= \"max-width:90%; height:auto;\" > (distributed (r-2) )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r%5E2-29r-18=-11r%2B22 ALIGN=MIDDLE  ALT=\"10r%5E2-29r-18=-11r%2B22\"   DISABLED_style= \"max-width:90%; height:auto;\" > (combined like terms \"r\")<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10r%5E2-18r-40=0 ALIGN=MIDDLE  ALT=\"10r%5E2-18r-40=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (added 11r, subtracted 22 from both sides)\r<BR>\n" );
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document.write( "That bad boy is a quadratic equation, and I can think of no better tool than the quadratic formula.  Hopefully you are already familiar with the quadratic formula, because instead of showing it, I just plugged it into the handy dandy solver (results below).  It gave me 2 values, but one of them was negative.  How can the boat go a negative rate?  That makes no sense in this problem.  We can throw that out.  That leaves the positive answer, 3.09, as the rate.\r<BR>\n" );
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document.write( "If you plug that in, you'll see it works.  Also, it took the boat 8.23 hours upstream, and 1.77 hours downstream.  (Holy cow!  It would have been MUCH faster to walk!)\r<BR>\n" );
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document.write( "<hr />For tutoring, please visit <a  rel=nofollow HREF=\"http://www.mathlocker.com\"><IMG STYLE=\"max-width:80%;\" src=\" http://www.mathlocker.com/images/math_locker_logo.jpg\" /></a><hr /><br />\r<BR>\n" );
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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=10x%5E2%2B-18x%2B-40+=+0 ALIGN=MIDDLE  ALT=\"10x%5E2%2B-18x%2B-40+=+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=%28-18%29%5E2-4%2A10%2A-40=1924 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-18%29%5E2-4%2A10%2A-40=1924\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1924 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--18%2B-sqrt%28+1924+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--18%2B-sqrt%28+1924+%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-%28-18%29%2Bsqrt%28+1924+%29%29%2F2%5C10+=+3.09317121994613 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-18%29%2Bsqrt%28+1924+%29%29%2F2%5C10+=+3.09317121994613\"   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-%28-18%29-sqrt%28+1924+%29%29%2F2%5C10+=+-1.29317121994613 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-18%29-sqrt%28+1924+%29%29%2F2%5C10+=+-1.29317121994613\"   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=10x%5E2%2B-18x%2B-40 ALIGN=MIDDLE  ALT=\"10x%5E2%2B-18x%2B-40\"   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=10x%5E2%2B-18x%2B-40+=+10%28x-3.09317121994613%29%2A%28x--1.29317121994613%29 ALIGN=MIDDLE  ALT=\"10x%5E2%2B-18x%2B-40+=+10%28x-3.09317121994613%29%2A%28x--1.29317121994613%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 3.09317121994613, -1.29317121994613.\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+10%2Ax%5E2%2B-18%2Ax%2B-40+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+10%2Ax%5E2%2B-18%2Ax%2B-40+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( " </SPAN>\n" );
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