Get the 46% 2,4-D from Tractor Supply AND a good backpack (4 gal) sprayer. Also, add a bit of surfactant to help the water stick better (some just use dish soap). Lastly, it's more effective if you spray while they're blooming.

I had dandelions in my pasture that must have been 30-40 years old, they had roots about 3/4 inch across and looked like long carrots.

What would Porter Rockwell do?