How to factorise any quadratic expression

Factorising quadratic equations

This method is used for the following factorable expression:

ax²+bx+c

Although this method is particularly useful with quadratic expressions with a≥2, it can be also used when a=1.

Given              ax²+bx+c

Find SUM=b and PRODUCT=ac

Find two numbers p and q, such that p+q=SUM and pq=PRODUCT

The smallest number (without considering the sign), say in this case p, goes into the following bracket:

(ax+p)

The largest number (without considering the sign), say in this case, q, goes into the other bracket:

(x+q/a)

Hence the factorised form is:

(ax+p)(x+q/a)

Further algebra could be used to "tidy" the expression

Example

6x² - 13x + 5

SUM = b = -13 and   PRODUCT = ac = 65 = 30

So p = -3 and q = -10 , as SUM= -3 -10 = -13   and PRODUCT= (-3)(-10) = 30

As p is the smallest number, this goes in (ax+p) = (6x-3)

And q being the largest, goes into (x+q/a) = (x-10/6)

Hence the factorised form is

 (6x-3)(x-10/6)

or neater (2x-1)(3x-5)