Sure! Here are the completed sentences along with explanations of why each answer fits:

1. My grandfather travelled a lot. He **could** speak five languages.
- **could** is used here to indicate ability in the past.

2. I looked everywhere for the book, but I **couldn’t** find it.
- **couldn’t** indicates inability to find the book.

3. They didn’t want to come with us at first, but we **managed to** persuade them.
- **managed to** suggests that despite difficulties, they were successful in persuading them.

4. Jessica had hurt her foot and **couldn’t** walk very well.
- **couldn’t** expresses her inability to walk well due to her injury.

5. There was a small fire in the kitchen, but fortunately I **managed to** put it out.
- **managed to** indicates that despite the potential danger, success was achieved in putting out the fire.

6. The walls were thin and I **could** hear people talking in the next room.
- **could** indicates the ability to hear despite the wall's thickness.

7. I ran my first marathon recently. It was very hard, but I **managed to** finish.
- **managed to** signifies that finishing was an accomplishment even though it was challenging.

8. My grandmother loved music. She **could** play the piano very well.
- **could** indicates her ability to play the piano.

9. We wanted to go to the concert, but we **couldn’t** get tickets.
- **couldn’t** indicates the inability to obtain tickets.

10. A girl fell into the river, but some people **managed to** pull her out. She’s all right now.
- **managed to** shows that despite a challenging situation, they succeeded in saving her.

Let me know if you need further explanation or assistance!