Problem 69 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (d+7)(d-7) $$

Step-by-Step Solution

Answer

The expression simplifies to \(d^2 - 49\).

Step 1: Recognize the Expression Type

The expression \((d+7)(d-7)\) is a product in the form of a difference of squares.\ This is because it has the form \((a+b)(a-b)\), where \(a = d\) and \(b = 7\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\). Applying this to \((d+7)(d-7)\), we substitute \(a = d\) and \(b = 7\) to obtain: \(d^2 - 7^2\).

Step 3: Simplify the Squares

Now, simplify \(7^2\) to get \(49\). Thus, the expression \(d^2 - 7^2\) becomes: \(d^2 - 49\).

Key Concepts

Understanding Binomial Products

Binomial products involve multiplying two binomials together. A binomial is an algebraic expression that has two terms. In the expression \((d+7)(d-7)\), each part within the parentheses is a binomial. A key takeaway here is learning how to multiply these expressions efficiently.

One approach to handle binomials is by using the formula for the difference of squares. This formula simplifies the multiplication of two binomials of the form \((a+b)(a-b)\). For \((d+7)(d-7)\), recognize that this structure fits perfectly. Thus, rather than expanding each term in a lengthy fashion, apply the difference of squares directly. It turns a complex multiplication into a straightforward subtraction problem.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the language through which we interpret many mathematical problems. In the difference of squares expression \((d+7)(d-7)\), 'd' is our variable, while the numbers like '7' are constants.

Typically, expressions are the stepping stones for solving equations. When dealing with algebraic expressions, identifying patterns such as binomial products or factoring can simplify your work significantly. This method of expression recognition is a skill that grows with practice. Also, algebraic expressions demonstrate how numbers relate under different operations, revealing the underlying rules that govern them.

Simplification Techniques in Algebra

One of the key skills in algebra is learning how to simplify expressions. Simplicity often reveals the essence of the problem at hand. In the case of the expression \((d+7)(d-7)\), using the difference of squares formula immediately simplifies the process.

To simplify expressions properly:

• Identify patterns (like \((a+b)(a-b)\)) for quick solutions.
• Perform operations such as squaring numbers accurately. For instance, calculating \(7^2\) as 49.
• Simplify results further, if possible, to reach the most compact form, such as turning \(d^2 - 49\) into its simplest expression without any further operations required.

Learning these techniques is essential as they not only save time but also provide a clearer understanding of the problem.