I agree with Leo Hyun-Sang Cho. But to add some detail, I often use 찾았어 when I want to say I've just found my lost item or things I was looking for such as (when you've find the lost item from somewhere) You can use ~에서 ~찾았어. or just ~찾았어. 드디어 반지를 찾았어. 냉장고에서 과일을 찾았어. And 찾아냈어/찾아냈다 is as same as 찾았어, but the difference it is more frequently used in the sentence about crime. 검찰은 ~에서 비리를 찾아냈다.

almost same. 찾다 1. find, look, search, hunt, seek, locate, recover 2. find, look, search, hunt, want, ask for 3. visit, call, look sb up, drop by, drop in 찾아내다 find, discover, locate '찾아내다' is composition verb.(합성동사) it means 찾다(find)+내다(discover).