Question

In 1–2 sentences, describe a scenario in which you might need to use the difference of two squares identity to find the product of two numbers. First describe the general mathematical conditions and explain how you would find x and a. Then provide a specific, real-world scenario.

Answer 1

The difference of two squares identity states that \( a^2 - b^2 = (a - b)(a + b) \) can be used to find the product of two numbers in situations where you want to calculate the square of a number minus the square of another. To apply this, you would identify \( x \) as the average of the two numbers and \( a \) as the amount each number differs from \( x \), allowing you to express the product as \( (x-a)(x+a) \).

For example, if you need to determine the product of 48 and 52, you can use the identity by recognizing that \( 48 = 50 - 2 \) and \( 52 = 50 + 2 \). Here, \( x = 50 \) and \( a = 2 \). Using the difference of squares, you calculate \( 48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496 \).