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 $(53-3)$ which is $50$ - a nice number to work with.
By using $b=3$ we get:

• $53^2~=~(53+3)(53-3)~+~3^2$
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.

• $a^2-b^2~=~(a+b)(a-b)$
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 re-written as:

• $a^2~=~(a+b)(a-b)~+~b^2$

Related:
AGentleIntroductionToTimeComplexity
CountableSet
TypesOfNumber
(none) CosineRule
DiophantineEquation
MagnitudeOfAComplexNumber
MagnitudeOfAVector
PolarRepresentationOfAComplexNumber
Pythagoras
RootTwoIsIrrational
Surd
SquareNumber PythagorasTheorem
RationalisingTheDenominator

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