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Recently, the MySQL package on Ubuntu was changed, so the user is no longer prompted to enter the password during installation. These instructions will help you set the password on Ubuntu 18.04, 18.10, 19.04, 19.10, 20.04, and others.

Step 1. Install MySQL server (if you haven’t already)

apt install -y mysql-server php-mysql

Step 2. Setup password authentication method

mysql -u root

USE mysql;
UPDATE user SET plugin='mysql_native_password' WHERE User='root';
FLUSH PRIVILEGES;
exit;

service mysql restart

Step 3. Set the root password

mysql_secure_installation

Enter current password for root (enter for none): Just press Enter
Set root password? [Y/n]: Y
New password: Enter password
Re-enter new password: Repeat password
Remove anonymous users? [Y/n]: Y
Disallow root login remotely? [Y/n]: Y
Remove test database and access to it? [Y/n]:  Y
Reload privilege tables now? [Y/n]:  Y

Step 4. Test the settings

mysql -u root -p

Try with no password and with correct password.