Solving 3x - 7 and 3x 7: A Comprehensive Guide to the Difference of Squares

Are you looking for a simple and efficient way to solve the algebraic expression 3x - 7 3x 7? You’ve come to the right place! Here, we will explore the solution step-by-step using the difference of squares formula. This method not only provides a straightforward answer but also helps you understand the underlying algebraic principles. Let’s dive in!

Overview of the Problem

The given problem is the expression 3x - 7 and 3x 7. To solve this, we can combine the two expressions by adding them together: 3x - 7 3x 7. Let’s break down this process and apply the difference of squares formula to find the solution.

Solution and Explanation

Step 1: Combine the Expressions

First, let's combine the given expressions 3x - 7 and 3x 7:

3x - 7 3x 7

Combining like terms, we get:

Step 2: Express the Combined Term as a Square

Now, we need to express this in a form that allows us to use the difference of squares formula. Notice that the original problem is actually the difference of two squares formula, (a^2 - b^2), applied to the terms 3x and 7:

(3x)^2 - (7)^2

Step 3: Apply the Difference of Squares Formula

The difference of squares formula states that (a^2 - b^2 (a b)(a - b)). Here, (a 3x) and (b 7). Applying the formula, we get:

(3x 7)(3x - 7)

Expanding this, we obtain:

9x^2 - 49

Thus, the solution to 3x - 7 and 3x 7 is 9x^2 - 49.

Additional Context: The Difference of Squares

The difference of squares is a fundamental concept in algebra. It is a special case of polynomial factorization and is expressed as:

a^2 - b^2 (a b)(a - b)

Understanding this formula helps in simplifying and solving a wide range of algebraic expressions and equations. By recognizing the structure of the problem, you can apply the difference of squares formula efficiently.

Summary: Applying the Difference of Squares

In this tutorial, we explored the problem of solving 3x - 7 and 3x 7. We combined the terms, recognized the problem as a difference of squares, and applied the formula to arrive at the solution 9x^2 - 49. The difference of squares formula is a powerful tool in algebra, and understanding it can greatly simplify your problem-solving process.

Conclusion

Whether you are a high school student or an advanced algebra enthusiast, the difference of squares is a crucial concept to master. Practice similar problems to reinforce your understanding and apply the formula confidently in various algebraic scenarios. Happy solving!