document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=843443>Question 843443</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me to find the answer. Find the length of a side of an equilateral triangle whose altitude is 3 feet shorter than a side. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #508213 by </B><A HREF=/tutors/aboutme.mpl?userid=LinnW><B>LinnW(1048)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=LinnW><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=LinnW><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=508213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Set s = length of a side.<BR>\n" );
document.write( "Bisecting one of the angles produces two triangles,<BR>\n" );
document.write( "each with a base of (1/2)s.  Since the height is 3<BR>\n" );
document.write( "less than a side length, height = s - 3<BR>\n" );
document.write( "By Pythagorean theorem<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2F2%29s%29%5E2+%2B+%28s+-+3%29%5E2+=+s%5E2 ALIGN=MIDDLE  ALT=\"%28%281%2F2%29s%29%5E2+%2B+%28s+-+3%29%5E2+=+s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F4%29s%5E2+%2B+s%5E2+-6s+%2B9+=+s%5E2 ALIGN=MIDDLE  ALT=\"%281%2F4%29s%5E2+%2B+s%5E2+-6s+%2B9+=+s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F4%29s%5E2+%2B+s%5E2+-6s+%2B9+=+s%5E2 ALIGN=MIDDLE  ALT=\"%281%2F4%29s%5E2+%2B+s%5E2+-6s+%2B9+=+s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2F4%29s%5E2+-6s+%2B9+=+s%5E2 ALIGN=MIDDLE  ALT=\"%285%2F4%29s%5E2+-6s+%2B9+=+s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "multiply each side by 4<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5s%5E2+-24s+%2B36+=+4s%5E2 ALIGN=MIDDLE  ALT=\"5s%5E2+-24s+%2B36+=+4s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "subtract 4s^2 from each side<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5E2+-24s+%2B36+=+0 ALIGN=MIDDLE  ALT=\"s%5E2+-24s+%2B36+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "There are two possible values for s.<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=1x%5E2%2B-24x%2B36+=+0 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-24x%2B36+=+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=%28-24%29%5E2-4%2A1%2A36=432 ALIGN=MIDDLE  ALT=\"b%5E2-4ac=%28-24%29%5E2-4%2A1%2A36=432\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Discriminant d=432 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--24%2B-sqrt%28+432+%29%29%2F2%5Ca ALIGN=MIDDLE  ALT=\"+x%5B12%5D+=+%28--24%2B-sqrt%28+432+%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-%28-24%29%2Bsqrt%28+432+%29%29%2F2%5C1+=+22.3923048454133 ALIGN=MIDDLE  ALT=\"x%5B1%5D+=+%28-%28-24%29%2Bsqrt%28+432+%29%29%2F2%5C1+=+22.3923048454133\"   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-%28-24%29-sqrt%28+432+%29%29%2F2%5C1+=+1.60769515458674 ALIGN=MIDDLE  ALT=\"x%5B2%5D+=+%28-%28-24%29-sqrt%28+432+%29%29%2F2%5C1+=+1.60769515458674\"   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-24x%2B36 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-24x%2B36\"   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-24x%2B36+=+1%28x-22.3923048454133%29%2A%28x-1.60769515458674%29 ALIGN=MIDDLE  ALT=\"1x%5E2%2B-24x%2B36+=+1%28x-22.3923048454133%29%2A%28x-1.60769515458674%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  Again, the answer is: 22.3923048454133, 1.60769515458674.\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-24%2Ax%2B36+%29 ALIGN=MIDDLE  ALT=\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-24%2Ax%2B36+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></TD></TR>\n" );
document.write( "</TABLE>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );