Lively discussion, thanks, Turbo, Don, Evo,
I tend to be on the conservative side and prefer to use natural oils, etc... on pests. That being said I have read this in at least 4 or 5 places and had almost as many tell me that short of taking a clean cutting from your plant and cloning it, there isn't much, organic or chemical that will kill 100% of them for good, as soon as their ideal proliferation conditions are met, surprise, surprise, they take hold again, then it is war to see if the plant can produce flowers/fruit before insect population(s) can overwhelm and kill it.
Last summer in June my plants outdoors had a large infestation of aphids (learned a lesson and will check for them every day), I stripped the leaves and soaked the plant in organic soap repeatedly, enough so that I got a very good harvest in September and October which I posted about here. If the root system is not compromised to the point that it can't support a healthy plant, drenching the roots can help. Even doing that, you can never get all of them. One article stated that if the outside temperatures were at or exceeded 42 Celsius or 107.6 Fahrenheit for a single day would wipe out the entire aphid population in the high temperature region. I recently posted a video where the hot chili pepper chef had his plants subjected to over 120 degrees Fahrenheit for two days and his peppers were mostly destroyed, he lost some leaves, and I would wager any little aphids and other insects of the same variety were history as well. The pepper plants seem to be able to tolerate higher temperatures than most pest varieties, they suffer damage, but not permanent. Peppers rule and insects drool I am afraid.
I suppose soon after I treat my big ghosts for aphids, I will have to re-pot them and check the roots to be sure the plants are worth saving. I will say this so far, I am impressed with their toughness and resiliency, not to mention the peppers and heat that they are capable of producing. Add the ingredient of hydroponics to the equation and I would wager things get tougher and more dicey to handle, for every solution there are more problems to deal with, complexity leads to more complexity and expands the range of possible solutions, and with it a hit or miss answers to the problems. Imagine if I had these problems on a large number of plants. Things can get expensive in a hurry with no guarantee of success.