Often, the simplest way to solve "ax² + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor.    So, while factoring may not always be successful, the Quadratic Formula can always find the solution.

The Quadratic Formula uses the "a", "b", and "c" from "ax² + bx + c", where "a", "b", and "c" are just numbers.   The Formula is derived from the process of completing the square, and is formally stated as:

 ax² + bx = –c   

x² + (b/a) x = –c / a

(b/2a)² = b² / 4a²

x² + (b/a) x + b² / 4a² = –c / a + b² / 4a² 

x² + (b/a) x + b² / 4a² = –4ac / 4a²  + b² / 4a² 

(x + b / 2a)² = (b² - 4ac) / 4a²

x + b / 2a =+ sqrt [(b² - 4ac) / 4a² ] = + sqrt (b² - 4ac)  / 2a

x = b / 2a + [sqrt (b² - 4ac)  / 2a]

x = [-b + sqrt (b² - 4ac)] / 2a