English company took the known flight path until the tracking device shut off. Using that information and known flight range, a possible flight area was made that stretched from the edge of the middle east (hence all the conspiracies of plane gonna be used for terror (even though that would make the plane fly into American military airspace)) to the southern parts of the Indian Ocean, extending out to about the Eastern coast of Africa.
From there, pings were detected from the blackbox coming from the Indian Ocean. Using the location data that could be gleamed from it, they established a likely crash area about the size of Texas, later reduced in size to about a large city. Now it's the process of finding a needle in a haystack, as they have to resort to sonar and submersible sweeps to find debris in that area. Not exactly easy.