document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=905428>Question 905428</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two cyclists start biking from the same location. Kate heads south at an average pace of 17 mph. An hour later, Nate heads west at an average pace of 14 mph. Which of the following could be used to determine how long after Kate starts riding it takes for them to be at least 35 miles apart? [Let t represent time.] </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #549200 by </B><A HREF=/tutors/aboutme.mpl?userid=ewatrrr><B>ewatrrr(24785)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ewatrrr><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=ewatrrr><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=549200\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Using Pythagorean Theorem<BR>\n" );
document.write( "(17t)^2 + (14(t-1))^2 = 35mi^2<BR>\n" );
document.write( "289t^2 + 196(t^2 - 2t + 1) = 1225mi<BR>\n" );
document.write( " 485t^2 -392t - 1029 = 0\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=485x%5E2%2B-392x%2B-1029+=+0 ALIGN=MIDDLE  ALT=\"485x%5E2%2B-392x%2B-1029+=+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=%28-392%29%5E2-4%2A485%2A-1029=2149924 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-392%29%5E2-4%2A485%2A-1029=2149924\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=2149924 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--392%2B-sqrt%28+2149924+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--392%2B-sqrt%28+2149924+%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-%28-392%29%2Bsqrt%28+2149924+%29%29%2F2%5C485+=+1.91573393180164 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-392%29%2Bsqrt%28+2149924+%29%29%2F2%5C485+=+1.91573393180164\"   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-%28-392%29-sqrt%28+2149924+%29%29%2F2%5C485+=+-1.10748650912123 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-392%29-sqrt%28+2149924+%29%29%2F2%5C485+=+-1.10748650912123\"   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=485x%5E2%2B-392x%2B-1029 ALIGN=MIDDLE  ALT=\"485x%5E2%2B-392x%2B-1029\"   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=485x%5E2%2B-392x%2B-1029+=+485%28x-1.91573393180164%29%2A%28x--1.10748650912123%29 ALIGN=MIDDLE  ALT=\"485x%5E2%2B-392x%2B-1029+=+485%28x-1.91573393180164%29%2A%28x--1.10748650912123%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 1.91573393180164, -1.10748650912123.\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+485%2Ax%5E2%2B-392%2Ax%2B-1029+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+485%2Ax%5E2%2B-392%2Ax%2B-1029+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );