Factoring
The factors of a quadratic equation will be (qx + r) and (sx+t). If the solutions are all integers, you may be able to quickly find q, r, s and t. The advantage of this method is that factoring can be very fast. The disadvantage is that factoring may not work; for instance, factoring will not find solutions that are not integers.
Completing the Square
Completing the square is a multistep process. The main idea is to convert the original equation into one of the form (x + a)^2 = b, where a and b are constants. The advantage of this method are that it always works and that completing the square gives some insight into how algebra works more generally. The disadvantage is that this method is complex.
Quadratic Formula
The quadratic formula is x = (-b +- (b*2 - 4ac)^.5))/2a. The advantages of this method are that the quadratic formula always works and is straightforward. The disadvantages are that the formula provides no insight and can become a rote technique.
Guessing
Sometimes, you can guess an approximate solution. Then, you can increase or decrease your guess, depending on whether the result from your first guess is too big or too small. The advantages of this method are that guessing can be very fast if you guess right, and can get an approximate answer quickly, if that is all you need. The disadvantage is that sometimes you won't be able to make a good guess .
