The difference of two squares identity states that \ (a^2 - b^2 = (a - b) (a + b) \). This formula can be used to find the product of two numbers in situations where you need to calculate the square of one number minus the square of another. Using this approach, you would identify \ (x \) as the average of the two numbers and \ (a \) as the difference between the two numbers, allowing you to formulate the product as $(x-a) (x+a) $. For example, if you need to figure out the product of 48 and 52, you can use the identity by noting that 48 = 50 - 2 and 52 = 50 + 2. Here, \ (x \) is equal to 50 and \ (a \) is equal to 2. As a result of using the difference of squares, you calculate \ (48 \times 52 = (50 - 2) (50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496 \).

Question

The difference of two squares identity states that \ (a^2 - b^2 = (a - b) (a + b) \). This formula can be used to find the product of two numbers in situations where you need to calculate the square of one number minus the square of another. Using this approach, you would identify \ (x \) as the average of the two numbers and \ (a \) as the difference between the two numbers, allowing you to formulate the product as $(x-a) (x+a) $. For example, if you need to figure out the product of 48 and 52, you can use the identity by noting that 48 = 50 - 2 and 52 = 50 + 2. Here, \ (x \) is equal to 50 and \ (a \) is equal to 2. As a result of using the difference of squares, you 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 states that $ a^2 - b^2 = (a - b)(a + b) $. This useful formula allows us to compute the product of two numbers by expressing it in terms of their squares. Specifically, we can represent two numbers as follows:

Let $ x $ be the average of the two numbers.
Let $ a $ be half the difference between the two numbers.

Using these definitions, the product of our two numbers can be formulated as $ (x-a)(x+a) $.

For example, to find the product of 48 and 52, we recognize that:

The average $ x $ is 50,
The difference $ a $ is 2, since 48 can be expressed as $ 50 - 2 $ and 52 as $ 50 + 2 $.

Now, we can apply the difference of squares identity:

\[ 48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 \]

Calculating further:

\[ 50^2 = 2500 \quad \text{and} \quad 2^2 = 4 \]

Thus, we have:

\[ 48 \times 52 = 2500 - 4 = 2496 \]

This method not only simplifies the calculation but also highlights the efficiency of the difference of squares identity in finding products of numbers that are symmetrically located around a central value.