Is one sacrifice sufficient for the entire family living under one roof?

To supplement Medi1Saif's answer, evidence that is cited by the Hanafis to support their point of view includes:

قل إن صلاتي ونسكي ومحياي ومماتي لله رب العالمين لا شريك له وبذلك أمرت وأنا أول المسلمين

Say, "Indeed, my prayer, my rites of sacrifice, my living and my dying are for Allah, Lord of the worlds. No partner has He. And this I have been commanded, and I am the first [among you] of the Muslims."

^{— Quran 6:162-163}

فصل لربك وانحر

So pray to your Lord and sacrifice

^{— Quran 108:2}

This implies an individual commandment similar to prayer with which it has been paired; just like prayer from one person in a home will not suffice for the rest, one sacrifice will not suffice for the rest.

They cite the hadith:

من كان له سعة ولم يضح فلا يقربن مصلانا

Whoever can afford it, but does not offer a sacrifice, let him not come near our prayer place.

^{— Sunan Ibn Majah}

This is firstly used to establish the obligation because there is a warning for not doing it.

Secondly a person who hasn't sacrificed but someone from his household has sacrificed will come under the same because of the apparent meaning. The hadith has conditioned the obligation only on one being able to afford it.

The Hanafis do consider it permissible to share others in the reward for one's sacrifice, as was done by the Prophet when he shared his family and the entire Ummah in the reward for his sacrifice. Though this does not waive the individual obligation for the sacrifice.

First note that the majority of scholars consider offering a sacrifice for 'Id al-Adha a sunnah. Only the Hanafis -as a fiqh school- consider it as wajib for each person who is able to offer a sacrifice (Based on a hadith which was compilled by imam Ahmad in his Musnad, ibn Majah in his Sunan, by al-Bayhaqi in his as-Sunan al-Kubra and his teacher al-Hakim al-Mustadrak and there qualified as Sahih by a-Dhahabi, see in here bulugh al-Maraam). Therefore imam Ahmad has a similar statement and this is also the choice of ibn Taymiyyah if the ability is given. I heard in a khotbah that imam Malik even considered it as best sunnah if a person was able to sacrifice a sheep on each member, but this is a recommendation.
Note that sharing the price of a sheep is not allowed in contrary to cows or camels. This means the person who will offer the sacrifice must pay it from his own money.

The majority follows the statement of the hadith of abu Ayub al-Ansari:

"We used to sacrifice one sheep, and a man sacrificed for himself and his family. Then later on people began to compete with each other and it became boasting."
(See for example in al-Muwatta' followed by imam Malik's point of view on sharing Camels etcs., see also in Sunan ibn Majah and Jami' at-Tirmdihi)

The Hanbali scholar al-Bahuti said refering to this hadith in his commentary on "Zad al-Mustanqi'":

(وتجزئ الشاة عن واحد) وأهل بيته وعياله (Source)
(And the sheep is sufficient for a single person) and his wives and kids(*)

(*) means even if he had a lot of them!

So according to the Hanafi view it is not enough if the person in charge of the family offers a sacrifice on behalf of the whole family under the same roof if there are others who would be able to offer a sacrifice too.

Note that when it comes to the interpretation of the hadith of abu Ayub the opinions divide into four views (One could say one of the maliki's and three different views in the shafi'i school of fiqh):

• The Malikis say there are three conditions for the validity: The person who would sacrifice on their behalf is living with them, in charge of them and a relative, be it father or husband or brother or son etc. (This view is the closest or that shared by hanbalis or salafis in their fatwas)
• 2nd opinion: People that have the same person which is in charge of them. Held by later scholars of the shafi'i school of fiqh. (This opinion is also accepted by salafis and hanbalis to some extent)
• 3rd opinion: all relatives of the person who would offer the sacrifice no matter if he is in charge of them or not.
• 4th opinion: Those who live with the person who would offer the sacrifice no matter if they are relatives or not. This view is held by some shafi'i scholars such as Shihab ad-Dyn ar-Ramli شهاب الدين الرملي while ibn Hajar al-Haythami strongly rejected this opinion.

Most fatwas I found say:

• if we have the case that you are married and live with your parents and rest of your family and have your own family you are in charge of. In this case it is strongly recommended to have your own sheep beside that of your parents, as you are in charge of an own family. This is also covered in the fatwa of the next example!
• It is allowed to share a sheep for both families of brothers if they share the meals together in the same house (ibn Bazz said about the same situation if both brothers are able to offer a sacrifice they should offer one for each family). (See for example islamqa #96741)
• On the other hand it is not considered as valid to share a sheep for example between father and son if the son is living in a separated home. (See for example islamqa #41766)

Note that the prophet () has sacrificed two sheep one for his family and one for those who are not able to sacrifice (Is it mandatory to mention person name for Qurbani?). And the prophet () actually had more than one wife living in more than one separate room and/or roof!

My primary source islamqa #160395.