Editing Difference of two squares (section)

==Proof==
=== 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.

=== Geometric proof ===
[[Image:Difference of two squares.svg|right|170px]]

The difference of two squares can also be illustrated geometrically as the difference of two square areas in a [[Plane (mathematics)|plane]]. In the diagram, the shaded part represents the difference between the areas of the two squares, i.e. <math>a^2 - b^2</math>.  The area of the shaded part can be found by adding the areas of the two rectangles; <math>a(a-b) + b(a-b)</math>, which can be factorized to <math>(a+b)(a-b)</math>.  Therefore, <math>a^2 - b^2 = (a+b)(a-b)</math>.

Another geometric proof proceeds as follows. We start with the figure shown in the first diagram below, a large square with a smaller square removed from it. The side of the entire square is a, and the side of the small removed square is b; thus, the area of the shaded region is <math>a^2-b^2</math>. A cut is made, splitting the region into two rectangular pieces, as shown in the second diagram. The larger piece, at the top, has width a and height a-b. The smaller piece, at the bottom, has width a-b and height b. Now the smaller piece can be detached, rotated, and placed to the right of the larger piece. In this new arrangement, shown in the last diagram below, the two pieces together form a rectangle, whose width is <math>a+b</math> and whose height is <math>a-b</math>. This rectangle's area is <math>(a+b)(a-b)</math>. Since this rectangle came from rearranging the original figure, it must have the same area as the original figure. Therefore, <math>a^2-b^2 = (a+b)(a-b)</math>.<ref>{{cite journal |last1=Slavit |first1=David |title=Revisiting a Difference of Squares |journal=Mathematics Teaching in the Middle School |date=2001 |volume=6 |issue=6 |page=381 |url=https://www.jstor.org/stable/41180978 |access-date=10 April 2025 |issn=1072-0839}}</ref>
[[Image:Difference of two squares geometric proof.png]]