ragingcheetah distro are you using . the new ff should be in YOUR distro software repo ( yum , apt-get,synaptic,...)
from the install instructions ( all quotes are copied from the instructions)
to repost that page
there is NO INSTALLER it is a binary download
bash: /root/firefox/firefox: No such file or directory
you DID NOT PUT it in the root home folder you put it in YOUR user home folder
also DO NOT RUN FIREFOX AS ROOT - that is a very bad idea .
WHERE did you save the tar.bz2 ( it should be in your user home folder )
1) extract the file to your home folder
tar xjf firefox-*.tar.bz2
"~" is your user home folder
Close Firefox if it's open.
run the firefox binary file
also WHAT distro are you using