A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a is not equal to zero.
The fastest way to solve a quadratic equation is through factoring, which involves inspection of the equation. If, upon inspection, the equation does not factor, you can try adding a value to both sides to complete the square. The quadratic formula is essentailly an already reduced version of competing the square. In my opinion the quadratic formula is consistently the most reliable to solve, but, when possible, I would like to factor first.