General Solutions to a Problem

[llamaprobe5]

I am trying to find a general rule for finding solutions to the problem ab-a-b=x. Solving for a or b, it is apparent that 2 and 2+x always work as solutions. Some values of x, however, such as as 50, have multiple solutions. Any suggestions as to how to solve for these solutions?

 

[gerben]

You can simplify the equation a bit:
ab-a-b = x
a(b-1)-b = x
substitute c = b-1
ac-c-1 = x
c(a-1) = x+1
substitute d=a-1
cd = x+1
so any two numbers whose product equals x+1 give a solution.

for x=50:
51 = 1*51 = 51*1 = 3*17 = 17*3 = (-1*)(-51) = (-51)*(-1) = (-3)*(-17) = (-17)*(-3)
so there are 8 solutions: (2, 52), (52, 2), (4, 18), (18, 4), (0,-50), (-50,0), (-2, -16), (-16, -2)