Question

How do you know when to use factoring or quadratic formula?

Answer 1

This is actually a very good question, but not one with a really definitive answer. The following is mostly some rules of thumb.

If the quadratic looks particularly " ugly " use the quadratic formula. Quadratics with coefficients that involve roots would be one example of "ugly".

After that (if the "ugly" rule doesn't apply):
Factoring is usually faster and less prone to arithmetic mistakes (if you are working by hand).
If the coefficient of `x^2` and the coefficient with no `x` element have relatively few factors, time invested in attempting to factor the quadratic is usually worthwhile.
Also if you know the source of the quadratic, you can sometimes guess if factoring is likely to be successful (for example if it is a simple mathematical model of a situation or a question developed by a friendly math teacher).

The Quadratic Formula takes longer
and you have to be careful with arithmetic
but it will give results when factoring won't work.
The quadratic formula may also demonstrate when no solution exists (something that is difficult to see with factoring).
Quadratics which arise from observed measurements and experimental results are more likely to need the use of the quadratic formula for solving.