Factorization little explanation ?

[lionely]

Alright in class, my teacher can factorize quadratics almost instantly.

I wanted to know if anyone can tell me how to do it his way...


Like if you had

5x^2 + 14x - 3
(x+3)(5x-1)

He writes that instantly, I kind of figured out in the first term, you put the sign that the b term has... then i get stuck.


By the way I know how to factorize the long way.. but I would like to learn this short way so I can work faster.

 

[Sourabh N]

If the quadratics are of the level of the example you gave, I'd say by practice. You work with them so much that you start seeing patterns right away.

For more complicated ones (9x^2 - 15x + 2, etc.), vedic mathematics has laid down some tricks.

 

[camjohn]

I think that with the level of factorization that you're describing, shortcuts are very possible.

1) In a given equation ax²+bx+c, play around with a and b to get c. What I mean by this is to either add them if c is positive (adding negative values is still adding keep in mind), or subtract them if c is negative. Unforunately, this almost always merely sends you in the right general direction for further analysis; for equations like the one you described, this will rarely give you the solution right away.
2) Play around with factors of a and b. Using the same methodology from step 1, this can give you the answer. For example, when I saw 5x²+14x-3, I figured out that since c was negative, a and b would have to be different values. Since c is 3, how can a or b be factored in such a way that the difference between the two equates to 3? Well, given that b is 14, b will have to be broken down. Since b is 14, it is likely that 5 and 3 could be used provided that one negative x value can be provided by the factorization. Since x has a high value, 5 must be aligned with x so that when the minus sign comes in the last parens with that beautiful little one, a high number still remains.
3) From there, the rest flows naturally.

Hope this helped give insight into some useful methods.