Looks good. My only comment is that if you're using mod 13, you should change the -4 to a 9.
I would like some help with this question:
Q: x^2+x+1=0 (mod 13)
I can't factorise the LHS as it stands, so I changed the 0 to 13, rearrange to give:
x^2+x-12=0 (mod 13)
(x+4)(x-3)=0 (mod 13)
so:
x=-4 or 3 (mod 13)
Does my method work or have I gone wrong somewhere? Pointers would be greatly appreciated.
Sort of.
You can always manipulate your constant term to a place where there are solutions within the range, like you did. It's a matter of making your determinant, , nonnegative.
However, it is possible that you will end up with non-whole number solutions, which may or may not be okay, depending on how strict you are with your modular definitions.
Your method is fine. You can plug your answers back into the original equation and see that they check. The quadratic formula works fine too, providing b^2 - 4ac has a square root in the field, and not everything does.
Completing the square works fine too, but not everything has a square root.
For example, consider x^2 + 4x + 6 = 0. If we try to complete the square we get first x^2 + 4x = -6 or x^2 + 4x + 4 = -6 +4 or (x+2)^2 = -2, or equivalently (x+2)^2 = 11. But there no integer whose square is 11 mod 13, so this equation has no solution.