Writing $axy + bx + cy + d$ in the form $a(x-\alpha)(y-\beta)+\gamma$

Question

I came across a problem in which we needed to rewrite an equation of the form: $$axy + bx + cy + d$$ in the form $$a(x-\alpha)(y-\beta)+\gamma$$ where $a,b,c,d$ are known constants and $\alpha,\beta,\gamma$ are constants to be found.

For example, we can write $$-4xyx+x+2y+2 \qquad\text{as}\qquad -4(x-0.5)(y-0.25)+2.5$$

I'm trying to generalise this factorisation so that I could do something similar irrespective of what the numbers are.

The form is quite similar to completing the square, and so I was wondering if perhaps this would require a similar approach.

Answer 1

First, divide by $a$ to get it in the form $xy+ux+vy+w $.

Then, note that $(x+v)(y+u) =xy+ux+vy+uv $, so $xy+ux+vy =(x+v)(y+u)-uv $.

This is the kind of algebraic manipulation that you need to get comfortable with.