3 Methods of Proof of the Difference of Squares Formula

The difference of squares is one of standard algebraic identities, which is used to factorize the terms like

a^2-b^2
(1)

That is, the difference of two squared terms is equal to the product of the sum of the two terms and the difference of the two terms. The difference of squares is commonly given as,

a^2-b^2=(a+b)(a-b)
(2)

The formula can be used for factoring polynomials with the square of one term minus the square of another. Before diving in the complex examples, let's find out different ways to prove the formula of the difference of squares, which can be done algebraically and geometrically.

Proof 1 - Algebraic method

Starting from the right-hand side of expression (2), apply the distributive law and cancel the middle terms,


\begin{aligned} (a+b)(a-b)&= a(a-b)+b(a-b) \\ &=a^2-ab+ba-b^2\\ &=a^2-b^2 \\ \end{aligned}

Proof 2 - Another Algebraic Method

Starting from the left-hand side (LHS) of expression (2), add and subtract the same term ab , which will not change the equation as,


ab-ab = 0


Then, we get,


\begin{aligned} a^2-b^2&= a^2-ab+ba-b^2\\ &=a(a-b)+b(a-b) \\ &=(a+b)(a-b) \\ \end{aligned}

Proof 3 - Geometrical Method

The difference of two squares can also be illustrated geometrically. The square immediately reminds us of the area of a square shape.


1. So lets construct 2 squares with their sides as a and b

2. Extend the top side of the larger square by the length of b to make another rectangle in green color with its adjacent side as a-b . In the mean time, a larger rectangle is formed plus the area of deep pink color.


So we get the area of pink colors (deep and light) is equal to the area of the larger rectangle.


\begin{aligned} a^2-b^2 &= a(a-b)+b(a-b) \\ &= (a+b)(a-b) \\ \end{aligned}

Steven Zheng Steven Zheng posted 4 months ago

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