The Prophet ﷺ said: “The one who does not perform sacrifice even though he can, should not join us in the Eid Prayer.”
(Musnad Ahmad, Ibn Majah)
This one hadith of Allah’s Messenger ﷺ is enough to show that sacrifice of an animal on the occasion of Eid al-Adha is of obligatory nature.
It is noteworthy that the command of sacrifice as contained in the Qur’ānic verses (22:36-37) is not for the pilgrims alone and that the performance of sacrifice is not confined to Makkah on the occasion of Hajj. It is a general command for all those Muslims who are well off. They have been enjoined to be grateful to Allah because He has subjected these animals for the good of all human beings. Therefore they are required to sacrifice the animals during these days so that they may spiritually join those who go to Makkah to perform Hajj.
In another hadith related by Anas, the Prophet ﷺ said: “The one who sacrificed an animal before the Eid Prayer, should offer another sacrifice; but the one who sacrificed his animal after the Eid Prayer, did the right thing and followed the way of the Muslims.” (Bukhari) It this connection, it is noteworthy that no Eid Prayer is held on the 10th of Zil-Hajj in Makkah: therefore the injunction was meant for all Muslims and not only for those performing Hajj at Makkah.
There are many authentic traditions to the effect that Prophet Muhammad ﷺ made sacrifice on this occasion, while he was personally at Madinah.