Here are some methods that can be used:
- Graphing
- Factoring.
- Completing the square.
- Using the quadratic formula.
- The Diagonal Sum Method. It quickly and directly gives the 2 real roots in the form of 2 fractions. It is a trial-and-error method, same as the factoring one, but it can reduce the number of permutations in half by using a Rule of Signs for real roots. In fact, it can be considered as a shortcut of the factoring method. When a= 1, it can give the 2 real roots quickly without factoring. Example. Solve x^2 - 39x + 108 = 0. The Rule of Signs indicates the 2 real roots are both positive. Write the factor-sets of c = 108. They are: (1, 108), (2, 54), (3, 36)...Stop! This sum is 36 + 3 = 39 = -b. The 2 real roots are 3 and 36. No needs for factoring! When a is not one, this new method selects all probable root-sets, in the form of 2 fractions. Then it applies a very simple formula to see which root-set is the answer. Usually, it requires less than 3 trials. If this new method fails, then you can positively state that this given quadratic equation can not be factored, and consequently the quadratic formula must be used. Please see book titled:"New methods for solving quadratic equations and inequalities" (Amazon e-book 2010). This book also introduces a new improved quadratic formula, in graphic form, that is easier to remember.