If you have never set a root password for MySQL server, the server does not require a password at all for connecting as root. To setup root password for first time, use mysqladmin command at shell prompt as follows:
$ mysqladmin -u root password NEWPASSWORD

However, if you want to change (or update) a root password, then you need to use the following command:
$ mysqladmin -u root -p'oldpassword' password newpass

For example, If the old password is abc, you can set the new password to 123456, enter:

$ mysqladmin -u root -p'abc' password '123456'

Change MySQL password for other users

To change a normal user password you need to type (let us assume you would like to change password for user vivek) the following command:
$ mysqladmin -u vivek -p oldpassword password newpass

Changing MySQL root user password using MySQL sql command

This is another method. MySQL stores username and passwords in user table inside MySQL database. You can directly update password using the following method to update or change password for user vivek:

1) Login to mysql server, type the following command at shell prompt:
$ mysql -u root -p

2) Use mysql database (type command at mysql> prompt):

mysql> use mysql;

3) Change password for user vivek, enter:

mysql> update user set password=PASSWORD("NEWPASSWORD") where User='vivek';

4) Finally, reload the privileges:

mysql> flush privileges;
mysql> quit
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