This second version forms the basis of a relatively quick way to
square integers, and that in turn gives us a technique for
multiplying integers in general.
To square a two digit number like $53,$ think of that as $a$ in the
above version.
We had a free choice for $b,$ and by using $b=3$
we get $(533)$ which is $50$  a nice number to work with. 

By using $b=3$ we get:
 $53^2~=~(53+3)(533)~+~3^2$
So that simplifies to
$53^2~=~56{\times}50~+~9.$
Now $56{\times}50$ is just $28{\times}100,$ which is $2800.$ So
that means $53^2~=~2800~+~9$ which is $2809.$ With a little practice
and knowledge of your times tables and number bonds, this can be done
quickly and easily in your head.


An identity that turns up repeatedly in algebra, and which needs to
be recognised in all its many disguises.
Note that this works in any
commutative algebra, which means it
applies not just to
integers, but also to
real numbers, and
complex numbers. It does
not apply to
matrices or
quaternions,
where multiplication is non
commutative.
It can also be rewritten as:
Related:
Local neighbourhood  D3