1. Difference of Two Squares
The figure shows a square of side a. a a b a b
A smaller square of side b is cut out from it.
Area of the remaining L-shape
= area of the larger square – area of the smaller square
= a2 – b2

2. a b
Area = a2 – b2
We can divide the remaining L-shape into two rectangles (I and II), and use them to form another rectangle. a b I I II II

3. What is the length of the new rectangle? a + bII I II
width?
length?
What is the length of the new rectangle?
a + b
What is the width of the new rectangle?
a – b
What is the area of the new rectangle?
(a + b)(a – b)

4. Can you set up an identity in terms of a and b?II
a – b
a + b
Area = a2 – b2
Area = (a + b)(a – b)
By comparing the areas, we can set up an important algebraic identity:
It is known as the identity of the difference of two squares.

5. Let’s prove this identity algebraically.
R.H.S. = (a + b)(a – b)
= (a + b)a – (a + b)b
= a2 + ba – ab – b2
= a2 – b2
L.H.S. = a2 – b2

6. Expand the following expressions by using the identity of the difference of two squares.
(h + 1)(h – 1)
(2x – 3y)(2x + 3y)

7. Follow-up question
Expand the following expressions by using the identity of the difference of two squares.
(a) (xy + 4)(xy – 4) (b) (3z – 8cd)(3z + 8cd)
Solution