There is no simple method of factorising a quadratic expression, but with a little practice it becomes easier. One systematic method, however, is as follows:
Factorise $ 12y^2 - 20y + 3 $
$ = 12y^2 - 18y - 2y + 3 $ (here the 20y has been split up into two numbers whose multiple is 36. 36 was chosen because this is the product of 12 and 3, the other two numbers).
The first two terms, 12y2 and -18y both divide by 6y, so 'take out' this factor of 6y.
$ 6y(2y - 3) - 2y + 3 $ (we can do this because 6y(2y - 3) is the same as 12y² - 18y) (see factorisation and simplification)
Now, make the last two expressions look like the expression in the bracket:
$ 6y(2y - 3) -1(2y - 3) $
The answer is $ (2y - 3)(6y - 1) $
How does Working out Quadratics by Factorization work?Edit
It works on the principle that if xy = 0. Either x, y or both must equal 0. When attempting to solve a quadratic equation in this method, you must always have 0 on the right hand side of the equation. Once a quadratic has been placed into the form (a +/- b) (a +/- c). One bracket can be removed and the equation can be solved as if it were linear -> a = 0-b and a = 0-c.