More eloquent idiom/expression for the phrase "cut it at the roots/source to prevent the problem from growing/escalating "

I suggest:

Nip it in the bud

Which means cut it off before it has a chance to grow.

McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

to put an end to something before it develops into something larger.

Wiktionary

(idiomatic) To stop something at an early stage.

If you see a bad habit begin to develop, try to nip it in the bud so that it does not become ingrained. Issues are easier to sort out the earlier they are addressed.

Allen's Dictionary of English Phrases

nip

nip something in the bud
to put an early stop to an activity, before it becomes established. A metaphor from the practice of removing the buds of plants to reduce their growth, and in early use often in the context of feelings and passions. The phrase emerges in Elizabethan drama from the 1560s on, first in the forms nip in the head and nip in the blade, and Shakespeare in 2 Henry VI (1591) III.i.89 has York tell King Henry 'Cold news for me, for I had hope of France, | As firmly as I hope for fertile England. | Thus are my blossoms blasted in the bud.' The earliest recorded occurrences of the phrase in the form now familiar are in plays of Fletcher and Dekker from the first decade of the 17th cent.

Lady Mary Wroth The Countesse of Mountgomeries Urania 1621
And all this was but melancholy, and truely that is enough to spoil any, so strangely it grows upon one, and so pleasing is the snare, as till it hath ruind one... This I have found and smarted with it; leave it then, and nip it in the bud, lest if blow to overthrow your life and happiness.

Dickens Pickwick Papers 1837
After great consternation had been excited in the mind of Mrs. Cluppins, by an attempt on the part of Tommy to recount how he had been cross-examined regarding the cupboard then in action, (which was fortunately nipped in the bud by his imbibing half a glass of the old crusted 'the wrong way', and thereby endangering his life for some seconds,) the party walked forth, in quest of a Hampstead stage.

You want to eliminate gossip in your group by heading it off at the pass. From Uncyclopedia

Head them off at the pass is a stock strategy of the heroes of Hollywood Western movies (or "oaters"), as they defeat the villains.

"The pass" is a mountain crossing, so narrow that they's horses done hafta cross it in single file. "Them" (usually rendered as "'em") would be the Bad Guys. What is never made clear is how the Good Guys have a way to get to the pass before the Bad Guys do. However, clearly, once they get to the pass, the Bad Guys could be encircled and trapped by the Good Guys. No oater is complete without a climactic race to "the pass," which the Good Guys win.

You may be able to head Jenna off at the pass by diverting her in some way before she starts to gossip. This might take more effort in the long run than eliminating her, but have other benefits, depending on what you mean by "eliminate".

We have to eliminate the tendency to gossip in our group of friends with a preemptive strike: Jenna must be eliminated.

1.pre-emptive strike - a surprise attack that is launched in order to prevent the enemy from doing it to you - see Fairfax

A very comparable idiom is "to cut the head off the snake"

It means to stop a larger problem by aiming at the source - often the leader, or a major culprit. The implication is that the rest of the problem will naturally die off without that source. It is sometimes used in military situations to mean targeting the head of an organization.

So, to use your example - "you want to deal with that gossiping group? Cut the head off the snake - Jenna must be let go".
(maybe don't say "Jenna must be eliminated", that might turn out a little strong!)

The longer version of the idiom is "Cut the head off a snake, and the body will die".

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Beginner's Yoruba

Abortion always provides a nice metaphor in sticky situations. Two examples: "the committee aborted the notion while it was still embryonic," & "the idea was killed in utero before it could grow."

abort

Oxford dictionaries