Should this sentence have a "nach" in it or is that not required (in this context) by "suchen"?
Warum sucht die Bundeswehr händeringend Personal, obwohl man dort viele Vergünstigungen bekommt, die es woanders nicht gibt?
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Sign up to join this communitySie sucht ihre Brille.
Sie sucht nach ihrer Brille.
The difference between those two is extremly subtle. The latter sentence puts a bit more focus on the action rather than the object.
AFAIK, "etwas suchen" und "nach etwas suchen" makes no difference in everyday use of german. I thought of a lot of constructs with "suchen" and the sentences could all be formed with "nach" without changing the meaning. Originally, there may have been a difference, but today those are pretty much interchangable. I guess "nach etwas suchen" is more used if you search for things. In english the difference is also not really clear, "to search something" and "to search for something".
EDIT: interestingly enough, most times i would translate "etwas suchen" to "to search for sth." and "nach etwas suchen" to "to search sth."
You could do either of both options. You can put nach in or you can omit it. You just have to be aware of where to put it. If you say "warum nach sucht die Bundeswehr..." its terribly wrong.
If you say instead "warum sucht die Bundeswehr händeringend nach Personal..." its perfectly fine.
In this case nach is completely optional as the sentence will be understood in both versions and there is no semantic difference to my knowledge (to be proven wrong by one of the other native speakers out here).