Ok, here are my results for getting rid of root aphids and ants.
Let me first explain that I had a severe infestation on my vegetable plants. So some of there may work for a mild case.
Ladybugs: They work well but they require a lot of prep. If you don't have plenty of ladybug attracting flowers in your yard already don't bother buying them. However, a research study found that mixing melted sugar with water will attract ladybugs and lacewings ans saw an average population increase of 300% -500%. If you already have a bad infestation in the soil it can prevent seeds from germinating and growing properly to already planted seeds it may not help. They will only eat what is on the surface and are likely to fly away.
Dish soap/water mixture: I have used this method on regular aphids with great results with in a 3 week period based on a once per week application. On root aphids this mixture didn't do too well. There was a noticeable reduction in the population with in the first few days after the first application but the population boomed again with in a week. The white 'eggs' that look like weird feathery roots remained unaffected and once hatched the cycle starts all over again. Application would need should be done 2x a week for as long as needed.
Dish soap/vegetable oil/water mixture: This has so far given me the fastest and most notable results. After two days I barely say any crawling (or flying) aphids and when I turned over the soil I didn't see any larva that was previously very obvious a day before application. However, I had a huge ant population boom followed by an increase in the aphid population (about 1/3 of the population before the first application.). This over all gave the best and fastest results and I highly recommend it.
Ortho Diaznon Ultra Insect Spray Multi- Purpose Insect Killer: Since the aphids were all over the yard I ended up giving in to a commercial product. This did an amazing job. All of my flower plants started a late bloom and had a nice growth spurt.
Other products that people swear by: diatomaceous earth, nematodes, and pyrethrum based insecticide.
Another very important part of eliminating aphids is to get rid of the ants that are probably farming them. Here are some methods that I have tried. The ants I am trying to get rid of are large black ants and small biting red fire ants. Both are extremely stubborn. All methods bellow, however, worked on small black ants.
Boiling Water: This seemed to extremely upset the ants and made them go into a panic mode but all it did was have the move the entrance of the ant hill one to three feet across for both type of ants.Warning: This method can harm plants so be careful where you pour.
Raid Spot Treatments: I tried just spraying the entrance with common Raid ant spray in a can but this didn't work well. Many large black ants died but they didn't even bother moving the ant hill. It worked better on the fire ants but didn't eliminate all the ants hills which only moved about a foot away.
Coffee Grounds: Both types of ants didn't even bat an eyelash at this method and didn't even bother moving their hill. They didn't even get upset over it. The red ants, however, avoided garden area where coffee grounds were on surface.
Boiling Water/Vegetable Oil: They were clearly upset but not enough to even move.
Create a Turf War: Shovel ants from one ant hill to another and vise versa. This actually worked! Since ants are territorial they are likely to start fighting each other. Two days after I did this all the red ant's hills were abandoned! A week later there were no sign of them. The large black ants, however, are still around but at least they don't bite me every chance they get.
Dig a hole & fill it up: (Used only on the large black ants) I did this purely out of frustration but it seemed to work at first but then they all showed up again in their same spot. Ant holes are essentially water proof because their tunnel system so I dug a hole right where their ant hill was. After that I filled it up with water at night (so there is less evaporation) for 3 days. WARNING: If you try this wear long pants and tuck them into your boots because the ants will attack. Also try stomping occasionally to shake them off. DOUBLE WARNING: Make sure that you're not digging over any pipes. In the U.S. you can call 811 and they will send over someone for free to mark where the utility lines are at.
Gasoline: (DO NOT TRY THIS) I really wanted to try to avoid this method for obvious reasons but my husband was getting tired of these ants (they even survived store=bought undiluted concentrated sprays last year) they were starting to become aggressive and the bites were extremely painful. He poured some gasoline into the hole (it wasn't near my vegetable garden or any other plants). And well, it works ... for 3 days until the ants that survived underground dug themselves out and formed a new any hill about a foot and half away.
Results: Ants win. The large unnamed black ants have had a huge reduction in population but nothing seems to get rid of them so I surrendered.