American Express Co. finally achieved a long-desired entry into the communications business yesterday, reaching an agreement with Warner Communications Inc., to buy a 50 percent interest in Warner's large cable television subsidiary.

Under a tentative accord approved by both corporations, Amex will pay $175 million in cash and short-term notes for its one-half interest in Warner Cable Corp. A jointly owned company, to be named Warner Amex Cabel Communications Inc., plans to establish a $250 million line of credit to finance future cable expansion.

Warner is a leader in the emerging cable television business, which is projected to explode with a home communications revolution in the next decade. The company has 140 cable systems in operation with some 650,000 subscribers in 29 states.

In addition, Warner operates a pay cable channel with 150,000 subscribers; a national pay cable network that features children's programming, with more than a million subscribers; and a unique two-way cable system called Qube, in Columbus, soon to be expanded to Houston and Cincinnati.

Last year, cable television companies had 14 million subscribers, about 19 percent of all television households. This was up sharply from a decade earlier, when cable systems connected only 3.6 million subscribers, or 6 percent of all TV homes.

New programming options available through multichannel cable systems as well as satellite transmission technology are expected to lead to continued cable growth and new pressure on the established TV networks.

Amex Chairman James Robinson III took note of this yesterday, asserting that the new firm will be part of a "major communications revolution in the U.S."

Warner and Amex will have equal representation on the board of Warner Amex Cable, and Warner Cable Preident Gustave Hauser will continue to serve in that capacity with the new joint venture, the companies announced.

Amex issues travel check and credit cards and is the nation's third-largest diversified financial services business, with assets of $15 billion and annual sales exceeding $4 billion. For some time, Amex has been seeking a merger or acquisition partner, and the field of communications has been a top priority.

An attempt by Robinson's company to acquire the McGraw-Hill publishing concern was defeated earlier this year when McGraw-Hill opposed the Amex bid in a bitter confrontation. In a Washington Post interview last June, Robinson said the board communications field remained a primary expansion goal. At the time, he expressed particular interest in cable and television, while ruling out a bid for newspaper publishing concerns.