Completing the square 2 variables

[p75213]

Homework Statement

Ax$^{2}$+By$^{2}$-Mxy

Capitals are constants.

The equation (one like it) is in a book I am reading where the author procedes to complete the square. He does it in one line without showing the procedure and I am completely baffled.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Well, what exactly is the problem? Do you know how to complete the square?
I will presume that you at least know that $(u+ v)^2= u^2+ 2uv+ v^2$.
Since you have $Ax^2$ and $By^2$ as $u^2$ and $v^2$, you must have $u= \sqrt{A}x$ and $v= \sqrt{B}y$ so that $2uv= 2\sqrt{AB}xy$.

That is, we can write
$$Ax^2+B^2- Mxy= Ax^2+ 2\sqrt{AB}xy+ By^2- Mxy- 2\sqrt{AB}xy$$
$$= (\sqrt{A}x+ \sqrt{B}y)^2- (Mxy+ 2\sqrt{AB}xy)$$

 

[p75213]

Here it is. I am pleased to say it all makes sense now.