Problem 100 - College Algebra

Question

Explain how to find the product of the sum and difference of two terms. Give an example with your explanation.

Step-by-Step Solution

Answer

The product of the sum and difference of two terms \(a\) and \(b\) (where \(a > b\)) is given by the formula \(a^2 - b^2\). An example using a=5, b=3 verifies this formula. The process involves first calculating the sum \((a + b)\) and difference \((a - b)\) of the terms, then multiplying them together to find the product.

Step 1: Understand the Terms and Identify their Sum and Difference

Firstly, let's consider two algebraic terms, let these be a and b. Their sum is given by \(a + b\), and difference by \(a - b\).

Step 2: Compute the Product of the Sum and Difference

Now we compute the product of the sum and difference of these terms, that is, we multiply \((a + b)\) with \((a - b)\). Using this formula: \((a + b)(a - b) = a^2 - b^2\), we see that the product is \(a^2 - b^2\).

Step 3: Verification with an example

To confirm the formula, let’s substitute a and b each with numerical values. Let a = 5 and b = 3. Hence, \((a + b)(a - b) = (5 + 3)(5 - 3) = 8 * 2 = 16\). And while calculating \(a^2 - b^2\) we have \(5^2 - 3^2 = 25 - 9 = 16\). Both show that they are equal, so their multiplication gives the same result.