Eradicate? No. Limit? Yes.
The only real effective approach is a holistic one that takes more intelligence and courage than most politicians show.
Leadership. Leadership needs to be provided to unite allied and other countries in an effort against terror. Amongst our allies there could be much better coordination, sharing, etc. We need to figure out a way to work with Russia and China, both have their own Islamic terror problems and our big players that could really help. The trick is dancing with them on terror even when we are at odds in other subjects. We are the superpower and that leadership falls on us. In different ways the Bush and Obama administrations have failed miserably at providing this leadership.
Money. You have to identify source of and stop the flow of money. It is not a peripheral issue but at the core of it. One of the failings, in my view, of Afghanistan was how the military essentially turned a blind eye to the drugs in Afghanistan which ended up being a huge source of funds for the Taliban. Of course, it is not as easy as just destroying crops etc because then you risk turning simple farmers trying to make a few bucks against you and then their whole family etc but you can't go around putting our fires (i.e. capturing weapons) and expect that to be effective. The smart thing is to remove the source of the fires and clear the brush (i.e. no money to buy weapons) before a fire can start.
Military engagement. I know this is the big issue here that some people freak out about. But, you CAN NOT allow Daesh and satellites to operate essentially almost freely in areas and expect that we can contain them or bomb them into submission. Al-Qaeda was decimated largely because we denied them a safe home base. The same must be done for Daesh and it's satellites. This does not always mean a large scale invasion and occupation- though at times it may. Increasingly we can count on these European countries that are being hit with terror to pony up and shoulder a larger burden supported by US logistics. For example, France has already shown it's ability to operate in Africa. With further assistance and encouragement, as part of a global strategy, this could be augmented and working along African nations then progress will be made. The air war will need to be continued and even further expanded but in areas special forces along with native or neighboring forces, our own troops, allied countries like France, etc will be needed.
Intelligence. We have to turn and infiltrate their organizations. The hard part is when they are small organizationally, typically they are full of true believers which makes it almost impossible to turn them and smaller organizations are hard to infiltrate normally. The larger they become, they easier it is to do both because not everyone playing the Jihadi is a true believer and you can play the Jihadi and be welcomed into the fold with some convincing.
There is more. I don't have the time right now to go into all of them in detail but education, propaganda (for lack of a better word), community relations within our borders with Arab/Muslim communities and protecting/assimilating those populations (not treating them like foreigners), encouraging refugees towards Turkey, Jordan, Saudi, etc and supporting those nations in hosting them, police/security improvements and economic/free trade development.