Describe three different ways to solve a quadratic equation. Describe the strengths and limitations of each method, and explain how you would decide which one to use to solve a particular problem.

There are 3 different ways to solve a quadratic.

Method one
The quadratic can be graphed. This is very useful to lead in the direction of a solution. Some graphing programs give you a very good hint at what your answer is going to be. Sometimes not. A graph is most useful when you are dealing with a problem like 
y = 3x + 2
y = x^2 - 5x + 6 where you want to know where the x and y values are the same for both equations. It may not give you an accurate answer, but it will tell you if the answers you got using other methods is correct.

It you need accurate answers and the the graph is not giving you good points then a graph is not that helpful.

Method Two
You could use the quadratic formula. It will give you an accurate answer but only if you don't make a mistake in the arithmetic. That's not a trivial consideration. If you are dealing with numbers like 3.444x^2 - 88.77x - 44.99 = 0 you have 2 major problems. One is not making a mistake and the other is knowing when to round. But in general, this is a fail safe method.

Method Three
You could try factoring. If it works, it's by far the quickest and in it's own way the most accurate. It is not always easy to see how numbers may factor. For instance, this quadratic can be factored, but is it worth finding out how?
y = 144x^2 + 560x - 221

Completing the Square
There are other methods that can be used. Completing the square is one of them. That gives the vertex which can also give the roots. It is prone to errors as well. If handled carefully, it is a good way to solve a quadratic. 
y = (x - 2)^2 - 4 will give you the vertex immediately and the roots if y is made to be 0. It will give you he y intercept by making x = 0 Those are the 5 major points of a quadratic is this is pretty useful. It's disadvantage is that people find it cumbersome to arrive at.

One Last Way
Lastly if all else fails, you could look for a program on the internet.