Looking for a Quadratic tip(s)

I found a quadratic calculator online, so to an extent this is a non-question. Also I have been doing this problem as a sort of hobby, so I'm much more interested in the process then the solution.

I have been trying to describe the intersection of two lines. After several pages of calculations I have ended up with a quadratic something like;

First off I assume that problems like this needed too be completed back in ye good ole days. And I am wondering if there is anything like the Rule of Sarrus, which is just a trick to make solving a matrix easier when doing it by hand.

Secondly I was wondering what kind of research has been done on rounding versus not rounding. Like say in the above quadratic, I just droped most of the decimal positions too make the calculations easier, what would this do to the two answers? (It seems like this sort of thing would be important in Engineering but I have no idea.)

Also, just to make sure I'm understanding this quadratic calculator correctly. It's output is a number that makes each binomial term equal to 0, after factoring?