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 when you want to calculate the square of one number minus the square of the other. To apply this, you can identify $x$ as the average of the two numbers and $a$ as the amount each number differs from $x$. Using this approach, you can express the product as $(x-a)(x+a)$.

Question

For example, to calculate the product of 48 and 52, you can use the identity $48 = 50 - 2$ and $52 = 50 + 2$. Here, $x = 50$ and $a = 2$. Using the difference of squares, you can calculate $48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496$.
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GPT-4o mini · Answer

The difference of squares identity, expressed as $a^2 - b^2 = (a - b)(a + b)$, provides a powerful tool for calculating the product of two numbers, especially when those numbers are close to each other. By leveraging this identity, you can simplify calculations significantly.

To apply this method, first identify $x$ as the average of the two numbers and $a$ as the half of their difference. This allows you to express the product of the two numbers as $(x - a)(x + a)$.

Let's take a practical example: calculating the product of 48 and 52.

Identify the average and difference: \[ x = \frac{48 + 52}{2} = 50, \] and \[ a = 52 - 50 = 2. \]
Apply the difference of squares identity: Now, rewrite the product: \[ 48 \times 52 = (50 - 2)(50 + 2). \] According to the identity, this can be simplified to: \[ 48 \times 52 = 50^2 - 2^2. \]
Calculate the squares: Evaluate the squares: \[ 50^2 = 2500, \] and \[ 2^2 = 4. \]
Find the final result: Subtract the two results: \[ 2500 - 4 = 2496. \]

Thus, using the difference of squares identity, we find that $48 \times 52 = 2496$. This method not only streamlines the calculation but also enhances accuracy by reducing the potential for error in more complicated multiplication scenarios.