document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=408890>Question 408890</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A batter hits a baseball and runs toward first base.  Lorenzo runs toward the ball to catch it before it hits the ground.  Ignoring wind resistance, the equation h(t)=-16t^2+42t+3.5 can be used to find h, the height of the ball in feet, using t, the time in seconds after it is hit.  How many seconds after the ball is hit does Lorenzo have before the ball hits the ground? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287968 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=287968\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> h(t)=-16t^2+42t+3.5<BR>\n" );
document.write( "set h(t) to zero and solve for t:<BR>\n" );
document.write( "0=-16t^2+42t+3.5<BR>\n" );
document.write( "apply the quadratic formula to give us:<BR>\n" );
document.write( "t = {-0.08, 2.71}<BR>\n" );
document.write( "we can throw out the negative solution leaving us with:<BR>\n" );
document.write( "t = 2.71 seconds<BR>\n" );
document.write( ".<BR>\n" );
document.write( "details of quadratic formula follows:<BR>\n" );
document.write( "<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=-16x%5E2%2B42x%2B3.5+=+0 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B42x%2B3.5+=+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=%2842%29%5E2-4%2A-16%2A3.5=1988 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%2842%29%5E2-4%2A-16%2A3.5=1988\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=1988 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+=+%28-42%2B-sqrt%28+1988+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28-42%2B-sqrt%28+1988+%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-%2842%29%2Bsqrt%28+1988+%29%29%2F2%5C-16+=+-0.0808435506004972 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%2842%29%2Bsqrt%28+1988+%29%29%2F2%5C-16+=+-0.0808435506004972\"   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-%2842%29-sqrt%28+1988+%29%29%2F2%5C-16+=+2.7058435506005 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%2842%29-sqrt%28+1988+%29%29%2F2%5C-16+=+2.7058435506005\"   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=-16x%5E2%2B42x%2B3.5 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B42x%2B3.5\"   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=-16x%5E2%2B42x%2B3.5+=+-16%28x--0.0808435506004972%29%2A%28x-2.7058435506005%29 ALIGN=MIDDLE  ALT=\"-16x%5E2%2B42x%2B3.5+=+-16%28x--0.0808435506004972%29%2A%28x-2.7058435506005%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: -0.0808435506004972, 2.7058435506005.\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+-16%2Ax%5E2%2B42%2Ax%2B3.5+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-16%2Ax%5E2%2B42%2Ax%2B3.5+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " = 0\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );