If $M = 3x^2 - 8xy + 9y^2 - 4x + 6y + 13$, where $x,y\in\mathbb R$, then $M$ must be:
a) positive $\qquad$b) negative $\qquad$c) $0 \qquad$ d) an integer
I somehow managed to figure it out by completing the square but in order to do so, it took me a lot of time and I'm not sure if every time I could solve such problems.
This whole expression can be written as: $$ 2(x - 2y)^2 + (x - 2)^2 + (y + 3)^2$$ which implies $M$ is positive.
My point is sometimes I'm lucky and I could group them in squares but other times not. Is there any particular technique/method which always works?
Secondly I also wanna know what you guys observe when completing the squares?