A broad spectrum medication is one that treats bacteria, fungus and protozoan infections. Anything that treats bacteria and fungus should do the job.
API General Cure contains Praziquantel and Metronidazole and probably won't do anything for this.
Praziquantel treats tapeworm and gill flukes.
Metronidazole treats internal bacteria and protozoan infections.
The filter should be left in the aquarium during treatment but remove the carbon from it so the carbon doesn't remove the medication. If you use a medication that wipes out beneficial filter bacteria, you can remove the filter media/ materials from the filter and put them in a bucket of tank water until treatment s over. Then put the media back in the filter. However, the filter should remain running during treatment so you don't get ammonia or nitrite readings and to help circulate the water. It's also a good idea to increase aeration/ surface turbulence when using medications or salt because they reduce the oxygen level in the water.
I don't know about the Tetra medication. Just follow the directions on the packet.
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Don't give fish salt baths. It stresses them out chasing them, catching them, and putting them in a container with salt water. Then catching them out and putting them back in the main tank, which has the pathogens in.
If you want to use salt, add it directly to the main tank so the entire tank gets salt. You can use salt with medications.
SALT
You can add rock salt (often sold as aquarium salt), swimming pool salt, or any non iodised salt (sodium chloride) to the aquarium at the dose rate of 1 heaped tablespoon per 20 litres (5 gallons) of water. If there is no improvement after 48 hours you can double that dose rate so there is 2 heaped tablespoons of salt per 20 litres.
Keep the salt level like this for 2 weeks.
The salt will not affect the beneficial filter bacteria, fish, plants, shrimp or snails.
After you use salt and the fish have recovered, you do a 10% water change each day for a week using only fresh water that has been dechlorinated. Then do a 20% water change each day for a week. Then you can do bigger water changes after that. This dilutes the salt out of the tank slowly so it doesn't harm the fish.
If you do water changes while using salt, you need to treat the new water with salt before adding it to the tank. This will keep the salt level stable in the tank and minimise stress on the fish.
When you first add salt, add the salt to a small bucket of tank water (2 litres or 1/2 gallon) and dissolve the salt. Then slowly pour the salt water into the tank near the filter outlet. Add the salt water over a couple of minutes.