Problem 100 - College Algebra Essentials

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 (a+b) and the difference (a-b) of two terms (a and b) is equal to the difference of the squares of the two terms, i.e., \(a^2 - b^2\). For example, with a=5 and b=3, both (5+3)(5-3) and \(5^2 - 3^2\) produce 16.

Step 1: Understand the Formula

First, become familiar with the formula \( (a+b)(a-b) = a^2 - b^2 \). This is also known as the difference of squares formula. The left-hand side of the formula means multiplying the sum 'a + b' and difference 'a - b' of two numbers 'a' and 'b'. The right-hand side of the formula means subtracting the square of second number 'b' from the square of the first number 'a'.

Step 2: Assist the Formula with An Example

To better understand the formula, an example is really effective. Let's consider a' = 5 and b' = 3. Now, putting these numbers into our formula, on the left side we get \( (5+3)(5-3) = 8*2 = 16 \), and on the right side we get \( 5^2 - 3^2 = 25 - 9 = 16 \). Both sides of the formula yielded the same result, 16, demonstrating that the formula is accurate.

Step 3: Conclusion

The product of the sum and the difference of two numbers is the difference of the squares of the two numbers. This algebraic formula is applicable for any real numbers.