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 \(48 \times 52 = (50 - 2)(50 + 2) = 50^2 - 2^2 = 2500 - 4 = 2496\).
make this better

Answer 1

The difference of squares identity provides an elegant way to compute the product of two numbers when their squares are involved. The identity states that \(a^2 - b^2 = (a - b)(a + b)\). This can be particularly useful for pairs of numbers that are symmetrically distributed around a central average.

To apply this method effectively, we can define \(x\) as the average of the two numbers and \(a\) as the difference between one of the numbers and the average. This leads us to express the product of the two numbers in the form \((x-a)(x+a)\).

For example, let's calculate the product of 48 and 52. Notice that we can express these numbers as \(48 = 50 - 2\) and \(52 = 50 + 2\). Here, we set \(x = 50\) (the average) and \(a = 2\) (the difference from the average).

Using the difference of squares identity, the product of 48 and 52 can be computed as follows:

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

Calculating these squares gives us:

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

Substituting back, we find:

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

Thus, using the difference of squares identity not only simplifies calculations but also reduces the potential for errors in arithmetic. Therefore, the product of 48 and 52 is \(2496\).