Editing Difference of two squares (section)

=== Algebraic proof ===
The [[mathematical proof|proof]] of the factorization identity is straightforward. Starting from the [[Sides of an equation|right-hand side]], apply the [[distributive law]] to get 
<math display=block>(a+b)(a-b) = a^2+ba-ab-b^2.</math>
By the [[commutative law]], the middle two terms cancel:
<math display=block>ba - ab = 0</math>
leaving<ref name="bbc-bitesize">{{cite web |title=Difference of two squares - Factorising an algebraic expression - National 5 Maths Revision |url=https://www.bbc.co.uk/bitesize/guides/zmvrd2p/revision/2 |website=BBC Bitesize |access-date=9 April 2025}}</ref>
<math display=block>(a+b)(a-b) = a^2-b^2.</math>
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 holds not only for numbers, but for elements of any [[commutative ring]]. Conversely, if this identity holds in a [[ring (mathematics)|ring]] {{mvar|R}} for all pairs of elements {{mvar|a}} and {{mvar|b}}, then {{mvar|R}} is commutative. To see this, apply the distributive law to the right-hand side of the equation and get
<math display=block>a^2 + ba - ab - b^2.</math>
For this to be equal to {{tmath|\textstyle a^2 - b^2}}, we must have
<math display=block>ba - ab = 0</math>
for all pairs {{mvar|a}}, {{mvar|b}}, so {{mvar|R}} is commutative.