Why do we use the quadratic formula? What are the benefits of solving a quadratic equation by the quadratic formula over the other methods we have?

The fact that taking a square root can give a positive or a negative answer can lead to the remarkable result that a quadratic equation has two solutions.

Step-by-step explanation:

You do not use the quadratic when the equation is factorable, and obvious. y = x^2 - 4x + 4 is factorable and obvious. y = x^2 - 2x + 1 is the same kind of equation. You can use the quadratic formula for it and it will give you an answer, but it is better just to learn how to factor it by sight. Here's an example of one that is factorable but it is not obvious. y=1364x^2+342x - 56 which can factor into (22x + 8) (62x - 7) but you'd never know it by just looking at it. The quadratic formula would solve the equation. You use the quadratic formula when the numbers are very awkward and they describe some problem in the "real" world. Throwing a spear of a javelin or a baseball does not have very nice numbers associated with them. Throw air resistance into it and you have a quadratic with really nasty numbers. The quadratic formula can be used (or part of it can be used) to see if the roots are real roots or if they are complex roots. You can't do a question like that one without using b^2 - 4*a*c