How do you factorise a quadratic when the coefficient of X^2 is not 1?

Let's take the expression: 2x²-7x+3

It's based on the general quadratic: ax²+bx+c, where a=2, b=(-7) and c= 3.

The rule of factorising is that we need to find such two numbers (k) and (m) that k*m = a*c and k+m = b.

In our case,

k*m=6 and

k+m = (-7)

You can either solve these simultaneous equations or you can use trial and error method to see that k=(-6) and m= (-1)

You can check it: (-6)+(-1) = (-7) and (-6)*(-1)=6

Now, re-write the initial quadratic so the coefficient (b) in it is written as (k+m). So, 2x²-7x+3 = 2x²-6x-x+3

Now, group the terms, and bring the common factor out:

2x²-6x-x+3= (2x²-6x)-(x-3) =

2x(x-3)-(x-3) = (2x-1)(x-3)

This your final factorised expression:

2x²-7x+3 = (2x-1)(x-3)