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 $.

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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.

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GPT-4o mini · Answer

The formula for the difference of squares states that $ a^2 - b^2 = (a - b)(a + b) $. This can be quite useful when you need to multiply two numbers that can be expressed in terms of squares. For example, to multiply 21 and 19, you can find their average, which is 20. Thus, you can rewrite 21 as $ 20 + 1 $ and 19 as $ 20 - 1 $. Here, if we assign $ a = 20 $ and $ b = 1 $, we can use the difference of squares identity: $ (20 - 1)(20 + 1) = 20^2 - 1^2 $. This results in $ 400 - 1 = 399 $.

To apply this in a real-world scenario, consider that you are designing a rectangular garden with an area of 399 square feet, and you already know that the length is 21 feet. By using the difference of squares trick, you can easily verify that the width would need to be 19 feet to achieve the specified area.