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So the electromagnetic force holds matter together, but in a quantum mechanical way. As for metallic bonds I have no clue how that works. The idea seems to be that a sea of electrons acts as a glue that holds together an array of nucleons. Not sure how Schrodinger's equation gets you that.
The strong force is indeed very strong, because it holds the nucleus of the atom together, even through electrostatic repulsion between positive electrons wants to pull it apart. The strong force only works in the nucleus of the atom. This is because it has a very short range.