Problem 41 - College Algebra Essentials

Question

$$\text { Factor the difference of two squares.}$$ $$36 x^{2}-49$$

Step-by-Step Solution

Answer

The factored form of the equation \(36x^2 - 49\) as the difference of two squares is \((6x - 7)(6x + 7)\)

Step 1: Identify the squares

The first step is to identify the perfect squares in the equation. We have \(36x^2\) and \(49\). These are both perfect squares, so we can signify them as \((6x)^2\) and \(7^2\) respectively.

Step 2: Apply the difference of squares formula

Following the formula \(a^2 - b^2 = (a - b)(a + b)\), substitute \(a\) with \(6x\) and \(b\) with \(7\). Therefore, we get \((6x - 7)(6x + 7)\).

Step 3: Writing the final answer

We hence have successfully factored the given equation. The equation \(36x^2 - 49\) can be expressed as difference of two squares, which gives us \((6x - 7)(6x + 7)\) as the final result.

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra. They consist of variables, numbers, and operations combined to represent a value. In the context of factoring, understanding algebraic expressions is crucial. For example, the difference of two squares is a special type of algebraic expression. This expression takes the form of \( a^2 - b^2 \), where each component \( a^2 \) and \( b^2 \) represents a perfect square, neatly tied together with a minus sign.

Using these expressions wisely allows for simplifying complicated problems into more manageable parts, as seen with the exercise \(36x^2 - 49\). By recognizing this pattern, we can apply specific methods like factoring to break down the expression into a product of simpler expressions, leading to easier solutions and better understanding of the relationships between these algebraic quantities.

Perfect Squares

Perfect squares are numbers or expressions that result from squaring an integer or a more complex algebraic expression. Recognizing perfect squares is essential when working with types of factoring, such as the difference of two squares. In our exercise \(36x^2 - 49\), we identify \(36x^2\) and \(49\) as perfect squares, related to \(6x\) and \(7\) respectively.

Understanding perfect squares provides a shortcut in the factoring process, enabling you to swiftly transform a problem into a solution. It's a pattern recognition skill that enhances your algebraic intuition. Once you've mastered the most common perfect squares, you can readily identify them even in more complex scenarios, laying the groundwork for more advanced mathematical problem-solving.

Factoring Polynomials

Factoring polynomials is a technique used to express a polynomial as the product of its factors, which are usually simpler polynomials. The difference of two squares, one of the simplest forms of factoring, capitalizes on a special pattern where two terms are squared and separated by a subtraction sign. This is particularly powerful because it simplifies expressions and, in some cases, solves equations. The steps provided in the solution employ this technique starting by identifying perfect squares and then applying the formula \(a^2 - b^2 = (a - b)(a + b)\).

This process transformed our original expression \(36x^2 - 49\) into the factored form \( (6x - 7)(6x + 7) \). Recognizing and utilizing such patterns not only speeds up the factoring process but also deepens your understanding of how algebraic expressions can be manipulated to reveal underlying relationships and solutions.