How I say with the correct grammar in English "This is the solution that I searched?"
I think the verb searched didn't is the correct conjugation of the verb
The word solution is a bit formal. Also "that I searched" is somewhat stilted, and, like solution, searched is a bit too formal.
All told, if you want to sound more natural, you would say something like this:
That's the answer I was looking for.
If you actually want to preserve the level of formality of the phrase in the question, the following would be more likely:
This is the solution I sought.
In English, you search somewhere or some place for something. The object you're trying to find goes with the preposition "for". In your sentence "solution" is what you were trying to find. Therefore the correct way to say it is
This is the solution that I searched for.
Compare with: This is the room that I searched.