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Goals from Omar Marmoush, Bernardo Silva and Nico Gonzalez lifted City up into third heading into Sunday’s final round of matches.
Mateo Kovacic’s red card 20 minutes from time could have given Bournemouth hope, but Lewis Cook was also sent off just minutes later for a lunge on Gonzalez.
There was further cause for celebration for City late on when Ballon d’Or winner Rodri made his first appearance since September as a substitute after suffering a serious knee injury.
City’s victory means a point when they travel to Fulham on the final day will be enough to secure a top-five finish thanks to their much superior goal difference compared to sixth-placed Aston Villa.
Just three days on from the disappointment of losing the FA Cup final to Crystal Palace, City had to raise themselves on an emotional night at the Etihad in De Bruyne’s final home match.
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