What is 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 [[Factorization|factored]] according to the [[Identity (mathematics)|identity]] :<math>a^2-b^2=(a+b)(a-b)</math> in [[elementary algebra]]. ==Proof== Starting from the [[Sides of an equation|left-hand side]], use the [[distributive law]] to get :<math>(a+b)(a-b)=a^2-ba-ab-b^2</math> By the [[commutative law]], the middle two terms [[Cancelling out|cancel]]: :<math>ba-ab=0</math> leaving :<math>(a+b)(a-b)=a^2-b^2</math> The result is one of the most commonly used identities in mathematics. Among other many uses, it gives simple proof of the [[AM–GM inequality]] in two variables. The proof holds in any [[commutative ring]]. However, if this identity holds in a [[ring (mathematics)|ring]] ''R'' for all pairs of elements ''a'' and ''b''. This means ''R'' is commutative. To see this, use the distributive law on the right-hand side of the equation and get :<math>a^2+ba-ab-b^2</math> For this equation to be equal to <math>a^2-b^2</math>, we must have :<math>ba-ab=0</math> for all pairs ''a'', and ''b''. So that ''R'' is commutative. {{math-stub}} [[Category:Algebra]]