The following information is from the Arizona Master Gardener Manual, which is published by University of Arizona Cooperative Extension.

Soil contains many of the nutrients required by plants.
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What is fertilizer?
Of the essential plant nutrients, nitrogen, phosphorus, and potassium – the primary nutrients – are most often included in fertilizers. The secondary nutrients − calcium, magnesium, and sulfur − are also needed in relatively large amounts but are usually not incorporated into fertilizers because in most soils they are present in sufficient quantities for healthy plant growth. Other nutrients, the micronutrients – boron, chlorine, copper, iron, manganese, molybdenum, nickel, and zinc – are needed in smaller amounts. If plants lack any of the essential elements, they can exhibit signs of nutrient deficiency. If the deficiency is severe enough, these symptoms can be visible.
Fertilizer grade
The fertilizer grade or analysis given on the package refers to the amount of nutrients present. All fertilizers are labeled with three numbers, giving the percentage by weight of nitrogen (N), phosphate (P₂O₅), and potash (K₂O). If we have a 100-pound bag of fertilizer labeled 10-10-10, it contains the equivalent of 10 pounds of N, 10 pounds of P₂O₅, and 10 pounds of K₂O. The fertilizer ratio indicates the relative proportions of nitrogen, phosphate, and potash contained in the fertilizer. The specific fertilizer ratio you will need depends on the soil nutrient level and crop needs. For example, a 1-1-1 ratio (10-10-10, 15-15-15, 20-20-20, etc.) is widely used at the time of lawn establishment, but established lawns generally respond better to fertilizer ratios high in nitrogen.
On fertilizer labels, the initials WIN and WSN stand for water-insoluble nitrogen and water-soluble nitrogen, respectively. The water-soluble nitrogen dissolves readily and is usually in very simple form, such as ammonium or nitrate nitrogen. Water-insoluble nitrogen consists of other nitrogen forms, such as organic nitrogen, and must be broken down into simpler forms before it can be used by plants. These are slow-release nitrogen sources and they deliver nitrogen at different rates according to the amount and kind of material in their composition.
Fertilizer containing N, P₂O₅, and K₂O is called complete fertilizer, while one that does not contain all three is an incomplete fertilizer. Examples of incomplete fertilizers are indicated in Table 2.7.
Special-purpose fertilizers
When fertilizer shopping, one can find fertilizers packaged for certain uses or types of plants such as citrus food, rose food, or azalea food. It is not possible to make a blanket statement that one fertilizer is best for every soil or for every area of the state. It is true that various plants use different nutrients at their own rates. However, plant roots are highly selective and do not require that nutrients are present in specific ratios.
Slow-release fertilizers need not be applied as frequently as other fertilizers, and higher amounts can be applied without danger of burning plants. Plants generally use the nitrogen in slow-release fertilizers more efficiently than nitrogen in other forms, since it is released over a longer period of time and in smaller quantities. Slow-release fertilizers, however, are more expensive than other types.
Organic fertilizers
Organic fertilizers are derived from the remains or by-products of once-living organisms. Cottonseed meal, blood meal, bone meal, hoof and horn meal, and animal manures are examples of organic fertilizers. When packaged as fertilizers, these products will have the fertilizer ratios stated on the labels. Some organic materials, such as composted manures, are sold as soil conditioners and do not have a nutrient guarantee.
Most organic fertilizers contain very low levels of nutrients relative to inorganic fertilizers. Therefore, they must be added at much higher rates. In general, organic fertilizers release nutrients over a fairly long period. A potential drawback is that they may not release enough nutrients when plants need them.
Compared to synthetic fertilizer formulations, organic fertilizers contain relatively low concentrations of actual nutrients, but they perform other important functions which synthetic formulations do not. They increase organic content of soil, improve physical structure of soil, and increase bacterial and fungal activity. Table 2.9 shows the approximate nutrient content of animal manures and suggested yearly rates of application per 1,000 square feet of garden area.
Fertilizers Combined with Pesticides
The major reason for buying a fertilizer combined with a pesticide is convenience. Fertilization and pest control can be accomplished in a single application, but it is more expensive than using separate materials. Also, the timing of fertilizer applications may not coincide with the appearance of a disease or an insect problem.
Fertilizer application
Soil texture is one factor affecting the frequency of fertilizer application. Sandy soils require more frequent applications of nitrogen and other nutrients than do soils with clay because nitrogen leaches out of sandy soils very easily. Other factors affecting application frequency include the type of crop − _species and variety − _the level of crop productivity required, frequency and amount of water applied, and type of fertilizer applied and its release rate.
Timing
Some crops are heavier feeders of particular nutrients than others. In general, plants that grow very rapidly require more nitrogen than slow-growing plants.
Perennials that have reached the desired size require much less fertilizer than young, rapidly-growing plants. Lush, leafy vegetables require frequent fertilization, whereas mature trees and shrubs perform nicely with one application every year or two.
Plants require the most fertilization when they are growing rapidly, so early season fertilization is critical. If an annual plant is stunted due to lack of early season nutrients, it will never catch up to well-fertilized counterparts, and yield will be compromised.
Application methods
Broadcasting: A recommended rate of fertilizer is evenly spread over the growing area and left to filter into the soil or it may be incorporated into the soil with a rototiller, spade, or with irrigation water.
Banding: Narrow bands of fertilizer are applied in furrows two to three inches from the garden seeds and 1 to 2 inches deeper than seeds. Banding can satisfy the needs of plants for phosphorus as the first roots develop. When fertilizers are broadcast and worked into soil, much of the phosphorus can be locked up by the soil and may not be available to the plant.
Starter solutions: Another way to satisfy the need for phosphorus when setting out transplants of tomatoes, eggplant, peppers, or cabbage is to use a liquid fertilizer high in phosphorus as a starter solution. Follow directions on the label.
Side dressing: Dry fertilizer is applied as a side dressing after plants are up and growing. Scatter fertilizer on both sides of the row 6 to 8 inches from the plants. Rake it into the soil and water thoroughly.
Foliar feeding: Foliar feeding is used primarily for micronutrients such as iron or zinc that can be bound by soil particles. Foliar-applied nutrients are sprayed onto the foliage and are absorbed through the leaf surface.
Improving soil structure
In some cases, coarse sand, vermiculite, and perlite are added to heavy clays to help improve the soil texture or structure. However, these inert materials can be expensive and large quantities are needed to do any good.
Compost, manures, and other organic amendments also are effective and economical for modifying soil structure. Organic matter is a great soil improver for both clay and sandy soils. Sources of organic matter include animal manures, compost, leaf mold, sawdust, and straw.
Green manure crops
Another source of inexpensive soil improvement is the cover crop. Green manures, or cover crops, are planted in the garden in the fall for incorporation in the spring. Any fast-growing annual plant that is easily killed will make a good green manure crop. Choices include annual ryegrass, oats, wheat, barley, vetch, or field peas. Cover cropping provides additional organic matter, holds nutrients that might have been lost over the winter, and helps reduce erosion and loss of topsoil. Legume cover crops can increase the amount of nitrogen in the soil and reduce fertilizer needs. Incorporate green manures at least two weeks before planting vegetables. They should not be allowed to go to seed.