Why is the default Search module not returning partial results?
Actually there's a quick way to set this up with the Search API and the Search API Database Search modules. No need for Solr.


Enable both the modules and go to the Search API configuration page
(admin/config/search/search_api)
Add a server and give it a recognizable name (such as "MySQL"), make sure it's enabled
Choose the following service class: Database service (normally the
only option. If you don't see it, make sure that the "Search API
Database Search" module is enabled).
Selecting this service class will provide you the option Search on
parts of a word (make sure you enable it). Now hit "Create
server". Next, go back to the Search API configuration page, and
add an index. Again give it a recognizable name, such as "Contentindex", and make sure it's enabled.
Select the server you recently created (in this example "MySQL").
Datasource options: check the content types you want to index (the
description of this field may be a bit confusing, but make no
mistake: you should check the content types you want to index, not
the ones you want to leave out).
You could check "Index items immediately", but that depends on the
size of your content. This option is alright for smaller websites.
In the other case: let cron take care of it. Hit "Create index".
Next, you want to enable the fields (in the fields tab) to that
very index, e.g.: contant type, title, status and author - and hit
"Save changes" (we'll take care of the related fields later).
Then you want to configure the filters (in the filter tab). Here
you can enable the bundle filter, if you want to finetune the
content types once again (probably won't need to).
Down below you'll find the highlight filter, to highlight the
search word in the results.
Now return to the fields tab and unfold the "Add related fields"
 tab. Add all the fields of which you want to get the content
 indexed, and hit Save Finally go to the "view" tab of the index,
 and index now all the items.


The last step is to create search results page and search block in Views, which only takes a few minutes.


Make sure the submodule Search Views is enabled and create a
view page which will show "Contentindex" (select the name you gave
to your index configuration instead of "content" which is selected
by default.
Add all the fields (make sure your format is "fields") that every
search result should consist of (title [link to content], body, …).
You'll notice that these field look like this Indexed Node: Title
Add an exposed filter: Search: Fulltext search; and select the
following in its settings: Contains any of these words, and use
as: Search keys.


You're done!

There's an excellent video tutorial on this partial search method without Solr, which also incorporates faceted search (which is not necessary for partial search, but in case you're interested…). If you're only looking for a partial search option, you can watch the video until 5:08 and after that skip to 6:45.