The Noble Quran teaches the method of prayer (salat), not Hadith.

The importance of Hadith is commonly emphasised by claiming that the Quran does not teach salaah even though it orders us to do so.  But here's what's overlooked.  Allah Almighty has  stated in His Final Message that the dictates given in the Quran and sent through our  Prophet Muhammad (SAAW) are the continuation of previous Divine Messages from the time of Prophet Abraham (pbuh) onward which the people did not bother to follow nor to preserve.   Commandments of salaah (prayer), zakkah (charity) fasting and pilgrimage (Hajj) were Divine commandments given to people through former messengers since ancient times.  Prophet Muhammad's mission and deliverence of the Quran was the final & complete repetition/confirmation of the Divine Message.  Hence, it would be correct to state that while Prophets Moses, Jesus and Muhamamd (peace on them all) delivered the Taurah, Bible and Quran respectively, all important and basic religious dictates i.e. salaah, charity and fasting came from the time of Prophet Abraham (peace on him), generation after generation.

Salaah was already taught to the people since the time of Prophet Abraham in the Arabian peninsula, but with the passage of time these people and their descendents deviated from the path of Allah.  They gradually turned away from the ideology of monotheism and embraced polytheism.

The 4 compulsory rules of salaah  elucidated by Allah in the Quran as commandments and also while narrating the events of former Prophets are ruku and sajoud (that is, bowing and prostrating), recitation of Surah Fatiha mentioned as "seven oft repeated verses," and the recitation of whichever Quranic verses we can memorize or remember.  These practices were passed on from the time of Prophet Abraham.

It should also be remembered that during the 7th century, before the advent of Islam, the pagan Arabs offered their prayers with much the same physical gestures as we do today, but the only big difference was that their prayer was not directed to Allah Almighty.   Instead, as polytheists, they offered their ruku and sajoud to their idols which were placed in the Kaaba at Makkah, and their concept of reaching out to Allah was through these idols.  This was the big mistake and corruption brought about by the people over the generations by which they ruined the monotheistic Faith as practiced by Prophet Abraham.

The purpose of the Quran regarding salaah (or prayer) was not to teach its method because the method was already known to the people.  But the purpose of the Quran regarding salaah was to correct its ideology, that is, salaah must be offered and directed ONLY to Allah alone.

As Shaikh Saad al-Humayd mentions in his article Did the Arabs know about Allah before Al-Islam that if we read the Quran carefully, we will realize that the pagan Arabs also believed in Allah Almighty, but they committed shirk because they attributed partners to Allah.

Hence, the method of salaah has been an ongoing process following the lineage from Prophets Abraham to Muhammad (peace on them).

Kindly check the following verses of the Noble Quraan for clarification and direct reference / evidence of the above analysis.

And they say: Be Jews or Christians, then ye will be rightly guided. Say (unto them, O Muhammad): Nay, but (we follow) the religion of Abraham, the upright, and he was not of the idolaters. (2:135)

Abraham was not a Jew, nor yet a Christian; but he was an upright man who had surrendered (to Allah), and he was not of the idolaters.
 Lo! those of mankind who have the best claim to Abraham are those who followed him, and this Prophet and those who believe (with him); and Allah is the Protecting Guardian of the believers. (3:67-68)

And strive for Allah with the endeavour which is His right. He hath chosen you and hath not laid upon you in religion any hardship; the faith of your father Abraham (is yours). He hath named you Muslims of old time and in this (Scripture), that the messenger may be a witness against you, and that ye may be witnesses against mankind. So establish worship, pay the poor-due, and hold fast to Allah. He is your Protecting Friend. A blessed Patron and a blessed Helper! (22:78)

And afterward We inspired thee (Muhammad, saying): Follow the religion of Abraham, as one by nature upright. He was not of the idolaters.  (16:123)

On prostrating (sajoud)

"Those were some of the prophets on whom Allah bestowed His Grace of the posterity of Adam, and of those whom We carried (in the Ark) with Noah, and of the posterity of Ibrahim and Israel - of those whom We guided and chose; whenever the Signs of Allah Most Gracious were rehearsed to them, they would fall down in prostrate adoration and in tears." (19:58)

"And among His Signs are the Night and the Day, and the Sun and the Moon. Don't prostrate yourselves to the Sun and the Moon, but prostrate to Allah, who created them if it is Him you wish to worship. If however, they are arrogant, those who are in the presence of your Lord celebrate His praises by night and by day and they never flag." (41:37-38)

"Do you then wonder at this recital? And will you laugh and not weep, wasting your time in vanities? Rather fall down in prostration to Allah and worship Him (alone)!"  (53:62)

"O you who believe! Bow down, prostrate yourselves, and worship your Lord; and do good so that you may prosper." (22:77)

"When it is said to them: 'Prostrate yourselves to the Most Gracious', they say: 'What is the Most Gracious? Shall we prostrate ourselves to that which you command us?' And it increases their flight from the Truth." (25:60)


THE METHOD OF SALAAT AS IN THE NOBLE QURAN:

A careful reading of the Noble Quran makes it clear that prayer offered to Almighty Allah by different people around the world does not necessarily require an identical procedure or language in order to be accepted by Him.  What's absolutely necessary is the concept of strict monotheism with sincerity and depth of one's devotion toward Allah only and NO other deity.  The only 4 rules of salaat mentioned in the Quran, which are:
(1) bowing (ruku)
(2) prostrating  (sajood)
(3) Recitation of Surah Fatiha mentioned in the Quran as "seven of the oft repeated (verses),"
"We have given thee seven of the oft-repeated (verses) and the great Qur'an."   (15:87)  Surah Al-Hijr.
(4) And reciting whichever Quranic verses we can easily remember.
"Recite, then, of the Qur'an that which is easy for you. .... So recite of it that which is easy (for you), and establish worship and pay the poor-due,..."  (73:20)   Surah Al-Muzzamil.

The above points make up the complete basic, mandatory method of salaat.  Anything else one might add by considering it to be compatible with Quranic principles is up to the individual.
 
Bowing and prostrating were done by all pious ones in the past, being significant signs of complete surrender to the will of Allah Almighty alone.   Therefore, the exact detailed method of salaah has not been highlighted in the Quran because obviously Allah left this task on His final messenger to teach the people.  So yes, the other details of salaah in addition to the 4 rules stated in the Quran were decided by the Prophet as he deemed best and thus demonstrated to his followers.  This was an ongoing and unwritten practice over generations (in accordance with the basics that prevailed during the time of Prophet Abraham) which the Hadith included (in bits and pieces) in its narrations some 200 years later.   Hence, the existence of Hadith can take no credit for "teaching salaah" as incorrectly claimed by many.   It should also be known that there's no single Hadith that gives a comprehensive description of the method of prayer.   Furthermore, it's also almost certain that many of the details added to salaah today as a part of its procedure have been invented by Hadith which was not said by the Prophet (sw).

By claiming the Hadith to be indispensible for teaching salaah, such people ought to realize the enormity of the violation of their claim.  They are, inadvertently or purposely, declaring the fallibility of the Quran in defence of the man-written Hadith by presuming the Quran to be incomplete when Allah Almighty has repeatedly said that His Book is complete (refer verse 5:3).  

This day are those who disbelieve in despair of (ever harming) your religion; so fear them not, fear Me! This day have I perfected your religion for you and completed My favor unto you, and have chosen for you as religion AL- ISLAM.  (5:3  Surah Al-Maidah)


P.S.

The standard comment from those so-called Muslims who do not want to accept the Noble Quran as a stand-alone Book has been that not just salaat, but all information on zakah, fasting and hajj also comes from Hadith.  And if you tell them that you don't accept Hadith, the common question they shoot at you is, "then how did you learn to say your prayers and the rules of fasting, charity and hajj?"  Well, that myth about Hadith teaching salaat (prayer) has already been fully clarified above.  Now, concerning the complete information on fasting, zakah and hajj can be found in the following links:

-  The Noble Quran on Fasting
-  The Noble Quran on Zakah
-  Quranic Information on Hajj