Hi & Welcome to Linux Questions:-
How do I expand the root partition?
Download g-parted live and use that. (one method)
Before you do you may want to list(using the fdisk -l command) all of the partitions on your machine so that you don't make a mistake in resizing/allocating partitions.
Why didn't the people who distributed Fedora 20 take care of this problem?
During your Fedora 20 installation the installer (Anaconda) should of allocated appropriate room for your root partition, home and swap. There should of been the option to manually partition if you didn't like what Anacoda was about to write to disk.
Is Fedora the only operating system on your pc?