How can you win the Queen and Rook vs Queen endgame?


Bruh, How do you lose with 1 Queen and 1 rook together?
Because they have a queen too
Don't expect a simple answer. These endgames are in the range of the endgame tablebase and it says that checkmating takes over 20 full moves. There are probably many variations which you can't all memorize. Best to trust on your common sense and your visualizations to find a safe place for your king to hide from the white checks. Once you succeed for a single instance you can probably checkmate the white king with your heavy artillery in a streak of checks. Or exchange the queens, or capture the white queen. There is always a way!

In these kinds of positions, You can just start giving checks and eventually you will either accidentally mate them or force them to trade queens.
- Keep giving checks, try to make them move the king into the same square colour as the queen, and look for forks via triangulation that would force them to have to trade their queen with your queen.
- Once the queens are off the board, just mate with the rook as a a king + rook vs lone king.
- Another alternative is to keep giving checks and try to get the king in front of the queen, give it a check so that it has to move, and you win the queen without trading.
It all depends on the position tbh.
How to win Q+R+K vs Q+K
First, try to trade queens, if your opponent accept it, then you will win the game with a rook, if your opponent declines, then try to force a trade of queens by a long fork ( i mean the king didn't defend the forking square) and you will win the game
Yes, the key concept is to exchange queens to create a KR v K endgame. To force the exchange, Black's Q must defend White's check while giving check; logical square is g5. Concept is easy, implementing it will take about 18 moves, then an equal number to mate; assuming you avoid 3-position draw and still have time on clock. A draw is your most likely result in the given position with white to move.
The key position to post was probably 5-8 moves earlier.