A furrow must be dug around the root until the lower part is exposed, then a dog is tied to it, after which the person tying the dog must get away. The dog endeavors to follow him, and so easily pulls up the root, but suddenly dies instead of the master. After this the root can be handled without fear.
(Credit goes to Wikipedia)
What can this root plant be used for? Well, some say that this can be a material used in creating a Homunculus. Also, according to the Harry Potter books (if you've read Chamber of Secrets), the Mandrake can be used to make a potion to heal petrification from a Basilisk's stoney gaze, which might be worth killing a dog, but I'd stay away from animal cruelty and just go with some really good ear plugs when taking out a Mandrake.
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