Problem 38 - Algebra 1: Concepts and Skills

Question

$$ (x+6)(x-6) $$

Step-by-Step Solution

Answer

The solution to the problem is \(x^2 - 36\).

Step 1: Identify the terms

The terms in the binomial expression are \(x + 6\) and \(x - 6\). Here, \(x\) is equivalent to \(a\) in our formula, and \(6\) is equivalent to \(b\).

Step 2: Apply the Difference of Squares Formula

Substitute \(a\) and \(b\) into the formula \(a^2 - b^2\). Here \(a\) is \(x\) and \(b\) is \(6\). Doing so gives: \(x^2 - 6^2\).

Step 3: Simplify the Expression

Finally, simplify \(x^2 - 6^2\) to get \(x^2 - 36\).

Key Concepts

Binomial Expressions

A binomial expression is a type of polynomial that contains exactly two terms. Think of it as an algebraic expression separated by a plus or minus sign. These terms can be variables, constants, or a combination of both, like in our example

• \((x+6)\)
• \((x-6)\)

Binomials are fundamental in algebra because they serve as building blocks for creating more complex expressions. Understanding binomials helps in simplifying algebraic equations and performing operations like adding, subtracting, multiplying, and factoring them. The key to working with binomials is recognizing their structure and how they relate to core algebraic operations.

Factoring

Factoring is the process of breaking down an algebraic expression into simpler parts, known as factors, that multiply together to give the original expression. In the context of the exercise, we are dealing with the difference of squares. The difference of squares is a specific pattern where you have two terms squared and subtracted from each other, like so:

• The pattern is expressed as \( a^2 - b^2 = (a + b)(a - b) \).
• In the example \((x+6)(x-6)\), the expression represents the factors of the difference of squares: \( x^2 - 6^2 \).

Recognizing this pattern lets you factor expressions quickly and efficiently, turning multiplication into addition and subtraction, which are generally easier to handle. This simplifies complex algebraic expressions and solutions.

Algebraic Simplification

Algebraic simplification involves rewriting an expression in a simpler or more concise form, without changing its value. This process is essential for making equations more manageable and can aid in further operations like solving or comparing expressions.
In the step-by-step solution provided, after identifying and factoring the original binomial expression

• \((x+6)(x-6)\)

we applied the difference of squares formula to attain \( x^2 - 36 \).
This expression \( x^2 - 36 \) is a simplified version of the original binomials, as it no longer features multiplication and is free of parentheses. By simplifying algebraic expressions, you can see the underlying relationships between terms and prepare the expression for additional operations like evaluation or graphing.