Use when:
1. you are told to solve by factoring.
Such as: "Solve by factoring".
2. the quadratic is easily factorable.
Such as: x^{2}  4x  12 = 0
3. the quadratic is already factored.
Such as: (x + 5)(x  8) = 0
4. the constant term, c, is missing.
Such as: 3x^{2}  x = 0 
Use when:
1. you are told to solve by square root method. Such as: "Solve by square root method".
2. x^{2} is set equal to a numeric value.
Such as: x^{2} = 9 or x^{2} = 12
3. the middle term, bx, is missing.
Such as: 3x^{2}  15 = 0
4. you have the difference of two squares.
Such as: x^{2}  81 = 0 
Use when:
1. you are told to solve by completing the square.
Such as: "Solve by completing the square".
2. you are told to put the quadratic into vertex form, a(x  h)^{2} + k = 0, before solving. 
Use when:
1. you are told to use the quadratic formula.
Such as: "Solve by the quadratic formula".
2. factoring looks difficult, or you are having trouble finding the correct factors.
Such as: 10x^{2}  3x  4 = 0
3. the quadratic is not factorable.
Such as: x^{2}  6x + 2 = 0
4. the question asks for the answers to
ax^{2} + bx + c = 0 to be rounded.
Such as: 2x^{2} + 18x + 4 = 0
5. the question asks for the answers to be written in a+bi form.
Such as: x^{2}  6x + 2 = 0

If you are graphing to FIND the zeros of the equation, then you are using a graphing utility (calculator). Any other method would have you finding the zeros BEFORE you draw the graph.
(Yes, the vertex method would create a graph without knowing the roots, but most likely you would still not know the roots.)
Use when:
1. you have a graphing utility (calculator) with the capability of finding the decimal values (approximations) of noninteger (unfriendly) roots (zeros).
If the roots are not obviously appearing as integer values on the xaxis of the graph, then use the graphing calculator's capabilities to determine the decimal values of the roots (usually a "Calculate ZERO" option on the calculator).
Remember, if your graph does not cross the xaxis, you will be dealing with complex roots and you must use a different method to find those roots. Such a method will be discussed in Algebra 2. 