The difference of two squares identity, expressed as \( a^2 - b^2 = (a - b)(a + b) \), can be useful when you want to calculate the product of two numbers that can be represented as the squares of two other numbers. For instance, if you need to find the product of 21 and 19, you can express these numbers in terms of 20 (which is roughly their average): \( 21 = 20 + 1 \) and \( 19 = 20 - 1 \). In this case, \( a = 20 \) and \( b = 1 \), allowing you to use the identity \( (20 - 1)(20 + 1) = 20^2 - 1^2 = 400 - 1 = 399 \).
In a real-world scenario, imagine you are designing a rectangular garden that needs to have an area of 399 square feet, and you know the garden's length is 21 feet. Using the difference of squares, you can easily verify that the width must indeed be 19 feet to meet the area requirement.