Unable to set or change root password

I have followed all the steps using mysqld_safe or mysql_secure_installation to try to set (or reset) the root password. I have tried to set it to my preferred password, and to blank. I have tried simple user "root", "root@localhost", and "root@127.0.0.1" singly and in all combinations. The only response I ever get after performing all these steps is "Access denied" either using password YES or NO. This completely prevents me from using the software.