I am trying to find a way to factorise a non-monic quadratic using a system of linear equations. I know there exist various algorithms which can help with this, however I'm not fond of such methods.
I first started by creating a general equation with 4 unkown variables excluding x :
Now lets say I was given the quadratic $6x^2-19x+15$. Using the general formula I made, I end up with the simultaneous equations:
Obviously this system of equations cannot be solved as there must be one more equation. Is there a fourth relationship between the variables which I can add or are there variables which I can eliminate? Any comments or suggestions are much appreciated :)
I have tried to remove one variable by using the standard equation:$$a(x-b)(x-c)=ax^2+x(ab+ac)+abc$$ Now using the previous quadratic $6x^2-19x+15$ I end up with 3 equations: $$a=6$$$$ab+ac=-19$$$$abc=15$$ However after solving these by hand I end up with another quadratic equation similar to my original expression: $$6b^2+19b+15$$