Difference of two squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity
The proof of the factorization identity is straightforward. Starting from the left-hand side, apply the distributive law to get
By the commutative law, the middle two terms cancel:
leaving
The resulting identity is one of the most commonly used in mathematics. Among many uses, it gives a simple proof of the AM–GM inequality in two variables.
The proof just given indicates the scope of the identity in abstract algebra: it will hold in any commutative ring R.
Conversely, if this identity holds in a ring R for all pairs of elements a and b of the ring, then R is commutative. To see this, apply the distributive law to the right-hand side of the original equation and get
and for this to be equal to , we must have
for all pairs a, b of elements of R, so the ring R is commutative.
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