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The best time to pull dandelions is when they are 2 weeks old or younger. Pull them when the lawn is wet. I like using a dandelion puller. Use the tool to cut around the roots to loosen them. You want to pull the whole root. Any part of the root longer than 1 inch can grow back.
Use the tool to grab the root and pry it out using the soil. Pull it up and out with your other hand. You can also pour some vinegar into the hole to kill any part of the root that remains.