Problem 81 - Beginning and Intermediate Algebra

Question

Find the following special products. $$(x+6)(x-6)$$

Step-by-Step Solution

Answer

The short answer is: \((x+6)(x-6) = x^2 - 36\).

Step 1: Identify the Difference of Squares formula

The difference of squares formula states that for any two terms 'a' and 'b': \((a+b)(a-b) = a^2 - b^2\)

Step 2: Apply the Difference of Squares formula

In this case, we can see that the problem resembles the difference of squares formula with \(a = x\) and \(b = 6\). So, using the formula, we have: \[(x+6)(x-6) = x^2 - 6^2\]

Step 3: Calculate the result

Now we can calculate the result: \[x^2 - 6^2 = x^2 - 36\] Thus, the special product of the given binomials is: \[(x+6)(x-6) = x^2 - 36\].

Key Concepts

special products

In algebra, there are certain multiplication patterns known as special products. These patterns help simplify computations and make sense of certain types of expressions without expanding them fully.
A key special product is the **difference of squares**, which involves an expression of the form \((a+b)(a-b)\).
This product is special because it always simplifies to \(a^2-b^2\). This happens because the middle terms in the binomial multiplication cancel each other out.

• This principle can save time and reduce errors when calculating.
• It's particularly useful for quickly solving or simplifying expressions like \((x+6)(x-6)\).

Recognizing these special patterns will make solving algebraic expressions more manageable and intuitive.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They're a fundamental component of math, enabling representation of generalized mathematical relationships.
With algebraic expressions like \(x+6\) and \(x-6\), we can perform numerous operations such as addition, subtraction, and multiplication to form equations or **simplified results**.

• Each part of an expression is called a term; for example, in \(x-6\), both \(x\) and \(-6\) are terms.
• A special product like \((x+6)(x-6)\) showcases the power of these expressions to compress and communicate significant amounts of numerical information in a compact form.

Understanding algebraic expressions is essential for exploring more advanced mathematical concepts.

binomial multiplication

Binomial multiplication involves multiplying two algebraic expressions, each containing two terms. These binomials are usually shown in the format \((a+b)(a-b)\).
This operation can seem daunting, but by using special products, you can simplify things significantly. Here's how it works for our specific case:

• The multiplication of \((x+6)(x-6)\) is a typical example of the difference of squares.
• Instead of multiplying each term, we apply the shortcut formula \(a^2 - b^2\).

This shortcut, derived from binomial multiplication, highlights the efficiency of special algebraic formulas. It allows for **quick computation** and understanding of algebraic structures without detailed expansion. Such methods are a neat trick for every math student to have in their toolbox.