NORWALK, Conn. » Priceline.com Inc. plans to buy online travel-research company Kayak Software Corp. for $1.8 billion to expand its travel business.

Priceline will pay about $500 million in cash and $1.3 billion in stock and assumed options. The deal values Kayak at $40 a share, a 29 percent premium over its closing price Thursday.

Shares of Kayak — which also reported a 78 percent jump in third-quarter earnings — soared in after-hours trading while Priceline shares fell.

Kayak allows users to compare hundreds of travel sites when looking for flights, hotels and rental cars. It then sends the consumer to other websites to complete their purchase and earns fees on the referrals, although some bookings can be made directly on Kayak’s website and mobile applications. It also sells advertising.

Kayak was created by the same executives who helped launch other travel sites including Expedia, Travelocity and Orbitz. The company went public in July after delaying its offering more than a year while it waited for the market to strengthen.