Answer: All annuals and practically all perennial plants can be successfully grown in containers. The only things to keep in mind are to choose containers with adequate drainage holes, and to water as often as necessary to keep the soil moist but not soggy wet. Since plants in containers need to be watered more often, which leaches out nutrients, you'll need to feed your plants more frequently. I use a half-strength dilution of liquid fertilizer and feed my containerized plants every 2-3 weeks over the growing season. This supplies a constant source of nutrients without the fear of over-fertilizing. Use potting soil rather than garden soil in your containers - it drains quickly yet holds just the right amount of moisture to keep the plant roots happy. The only exceptions to growing in containers might be your carrots and corn. Carrots need really deep soil. You can grow them in containers, but you'll want very deep ones. Corn has a very extensive root system and should be grown in the ground.
Keep all your vegetables in a site that receives full sunshine all day long and they should flower and produce lots of fruit.
Best wishes with your garden!
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