document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=404418>Question 404418</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve algebraically  e^x = -x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #285821 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=285821\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It's quite clear that no <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=+0 ALIGN=MIDDLE  ALT=\"x+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > can satisfy the equation. (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex+%3E+0 ALIGN=MIDDLE  ALT=\"e%5Ex+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x+%3C=+0 ALIGN=MIDDLE  ALT=\"-x+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >).  Considering the graphs of both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > and -x, there is only one solution (or intersection point), and it lies between -1 and -1/2.<BR>\n" );
document.write( "(This is because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-1%29+-+1+%3C+0 ALIGN=MIDDLE  ALT=\"e%5E%28-1%29+-+1+%3C+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28-1%2F2%29+-+1%2F2+%3E+0 ALIGN=MIDDLE  ALT=\"e%5E%28-1%2F2%29+-+1%2F2+%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex++=+1+%2B+x+%2B+x%5E2%2F2 ALIGN=MIDDLE  ALT=\"e%5Ex++=+1+%2B+x+%2B+x%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for small values of x.<BR>\n" );
document.write( "So to get an approximate value for the root, solve\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x+=+1%2Bx+%2B+x%5E2%2F2 ALIGN=MIDDLE  ALT=\"-x+=+1%2Bx+%2B+x%5E2%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F2+%2B+2x+%2B+1++=0 ALIGN=MIDDLE  ALT=\"x%5E2%2F2+%2B+2x+%2B+1++=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  or <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+4x+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+4x+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-4+%2B-+sqrt%28+16-8+%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+sqrt%28+16-8+%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-4+%2B-+2sqrt%282%29%29%2F2+ ALIGN=MIDDLE  ALT=\"x+=+%28-4+%2B-+2sqrt%282%29%29%2F2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2+%2B-+sqrt%282%29%29+ ALIGN=MIDDLE  ALT=\"x+=+-2+%2B-+sqrt%282%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Eliminating the bottom value the answer is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-2+%2B+sqrt%282%29%29+ ALIGN=MIDDLE  ALT=\"x+=+-2+%2B+sqrt%282%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );