yes he defeated both of them, he defeated grendel by ripping off his arm them keeping it as proof

After Beowulf slays Grendel's mother he chops off the head of Grendel. He killed Grendel's mother with a magical sword that he used to stab her in the neck.

Beowulf kills Grendel, Grendel's mother, and a dragon in the epic poem Beowulf.

Beowulf kills Grendel's mother by using a sword on her.

In the epic poem "Beowulf," the narrator refers to Grendel's mother simply as "Grendel's mother." She is not given a personal name in the story.