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.

finish this piece: You could use the difference of two squares' identity in architecture to make sure windows are perfectly similar or spaced out, or for pathways like the ones in gardens, with separated pieces.

Answer 1

In situations where you need to calculate the area of rectangular plots or spaces that are subtracted from each other, such as finding the area of a larger square minus a smaller square within it, you can apply the difference of two squares identity, which states that \(a^2 - b^2 = (a + b)(a - b)\). In this case, \(x\) represents the dimensions of the squares, while \(a\) and \(b\) are half the dimensions of the larger and smaller squares, respectively.

For instance, if an architect is designing a large square window frame (10 ft by 10 ft) and wants to determine the area of the frame that needs to be carved out for a smaller square glass pane (6 ft by 6 ft), they can represent this as \(10^2 - 6^2\), using the difference of squares identity to quickly find that the area of the frame around the window is \( (10 + 6)(10 - 6) = 16 \times 4 = 64 \text{ sq ft} \). This can ensure that the frame not only fits perfectly around the glass but also adheres to aesthetic uniformity in the design.