Exhibit 3.51

Delaware	PAGE 1
The First State

     I, JEFFREY W. BULLOCK, SECRETARY OF STATE OF THE STATE OF DELAWARE, DO HEREBY CERTIFY THE ATTACHED ARE TRUE AND CORRECT COPIES OF ALL DOCUMENTS ON FILE OF “CASTLEVIEW HOSPITAL, LLC” AS RECEIVED AND FILED IN THIS OFFICE.

     THE FOLLOWING DOCUMENTS HAVE BEEN CERTIFIED:

     CERTIFICATE OF FORMATION, FILED THE NINTH DAY OF NOVEMBER, A.D. 1998, AT 9 O’CLOCK A.M.

     CERTIFICATE OF AMENDMENT, FILED THE TWENTY-SECOND DAY OF JANUARY, A.D. 2002, AT 10 O’CLOCK A.M.

      AND I DO HEREBY FURTHER CERTIFY THAT THE AFORESAID CERTIFICATES ARE THE ONLY CERTIFICATES ON RECORD OF THE AFORESAID LIMITED LIABILITY COMPANY, “CASTLEVIEW HOSPITAL, LLC”.

2964208     8100H		/s/ Jeffrey W. Bullock
110292307 You may verify this certificate online at corp. delaware.gov/authver.shtml		Jeffrey W. Bullock, Secretary of State AUTHENTICATION: 8619856 DATE: 03-14-11

STATE OF DELAWARE

SECRETARY OF STATE

DIVISION OF CORPORATIONS

FILED 09: 00 AM 11/09/1998

981429958 — 2964208

CERTIFICATE OF FORMATION
OF
CASTLEVIEW HOSPITAL, LLC

Under Section 18-201 of the
Delaware Limited Liability Company Act

     FIRST: The name of the limited liability company is Castleview Hospital, LLC (the “Company”).

     SECOND: The address of the registered office of the Company in the State of Delaware is 1013 Centre Road, Wilmington, Delaware 19805.

     THIRD: The name and address of the Company’s registered agent for service of process is Corporation Service Company, 1013 Centre Road, Wilmington, Delaware 19805.

     IN WITNESS WHEREOF, the undersigned has executed this Certificate of Formation as of November 5, 1998.

By:	/s/ John M. Franck II
	Name:	John M. Franck II
	Title:	Authorized Person

STATE OF DELAWARE

SECRETARY OF STATE

DIVISION OF CORPORATIONS

FILED 10: 00 AM 01/22/2002

020040862 — 2964208

CERTIFICATE OF AMENDMENT

OF

Castleview Hospital, LLC

     1. The name of the limited liability company is
          Castleview Hospital, LLC

     2. The Certificate of Formation of the limited liability company is hereby amended as follows:

The name and address of the registered agent is The Corporation Trust Company Corporation Trust Center, 1209 Orange Street, Wilmington, Delaware 19801.

     IN WITNESS WHEREOF, the undersigned has executed this Certificate of Amendment of Castleview Hospital, LLC this 15 day of January, 2002.

Castleview Hospital, LLC

/s/ William F. Carpenter III  

William F. Carpenter III, 

Title  Manager