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?
Best Answer
Without completing the square, you can also apply the following technique:
$$\begin{align} &3x^2 - 4x(2y+1)+ (9y^2 + 6y + 13-M)=0\\ \implies &\Delta_x=4(2y+1)^2-3(9y^2+6y+13-M)≥0\\ \implies &3M≥11y^2+2y+35\\ \implies &3M≥11 \left(y + \frac{1}{11}\right)^2 + \frac{384}{11}\\ \implies &3M≥\frac{384}{11}\\ \implies &M≥\frac{128}{11}>0.\end{align}$$