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 $.

Question

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.

Rewrite this in your own words

GPT-4o mini · Answer

The difference of two squares formula states that $ a^2 - b^2 = (a - b)(a + b) $. This can be particularly helpful when calculating the product of two numbers that can be represented as squares. For instance, if you want to multiply 21 and 19, you could relate these numbers to the number 20, which is their midpoint. Specifically, you can express 21 as $ 20 + 1 $ and 19 as $ 20 - 1 $. In this scenario, you set $ a = 20 $ and $ b = 1 $, allowing you to apply the identity: $ (20 - 1)(20 + 1) = 20^2 - 1^2 $. This simplifies to $ 400 - 1 = 399 $.

To put this in a practical context, imagine you are planning a rectangular garden that needs to cover an area of 399 square feet, and you know that the garden's length is 21 feet. By utilizing the difference of squares, you can confirm that the width must be 19 feet in order to achieve the desired area.