Expanding And Simplifying (3x - 2)(3x - 13) A Step-by-Step Guide

by ADMIN 65 views

In this article, we will delve into the process of expanding and simplifying the algebraic expression (3x - 2)(3x - 13). This type of problem is a fundamental concept in algebra, often encountered in various mathematical contexts, from solving quadratic equations to simplifying complex expressions. Expanding and simplifying expressions is a critical skill for students and professionals alike, enabling them to manipulate equations effectively and solve a wide range of problems. Understanding the distributive property and how to apply it correctly is essential for this process. By the end of this guide, you will have a clear understanding of how to expand and simplify this particular expression, along with the general principles that apply to similar problems. This knowledge will not only help you in your academic pursuits but also in practical applications where algebraic manipulation is necessary. The ability to simplify expressions accurately is a cornerstone of mathematical proficiency, and mastering this skill will significantly enhance your problem-solving abilities in mathematics and beyond. So, let's begin with a step-by-step approach to unlock the simplified form of the given expression.

Understanding the Distributive Property

The cornerstone of expanding expressions like (3x - 2)(3x - 13) is the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In simpler terms, it means that you multiply each term inside the parentheses by the term outside. When dealing with two binomials (expressions with two terms), we extend this property using the FOIL method, which stands for First, Outer, Inner, Last. This method ensures that we multiply each term in the first binomial by each term in the second binomial. Let’s break down the FOIL method in the context of our expression. First, we multiply the First terms of each binomial, then the Outer terms, followed by the Inner terms, and finally the Last terms. Applying this systematically helps us keep track of all the necessary multiplications and ensures that we don’t miss any terms. Understanding this foundational principle is crucial because it’s the key to correctly expanding any similar algebraic expression. The distributive property is not just a rule to memorize; it’s a logical principle that simplifies complex multiplication by breaking it down into smaller, more manageable steps. In the next section, we will apply the FOIL method to our specific expression, demonstrating how each step of the distributive property contributes to the final expanded form.

Step-by-Step Expansion Using the FOIL Method

Now, let’s apply the FOIL method to expand the expression (3x - 2)(3x - 13). This step-by-step approach will help you understand each multiplication and how they combine to form the expanded expression.

  1. First: Multiply the first terms in each binomial: (3x * 3x = 9x^2). This is the initial part of our expanded expression.
  2. Outer: Multiply the outer terms in the expression: (3x * -13 = -39x). This gives us the second term in our expansion.
  3. Inner: Multiply the inner terms: (-2 * 3x = -6x). This is the third term we need to include.
  4. Last: Multiply the last terms in each binomial: (-2 * -13 = 26). This completes the multiplication process.

So, applying the FOIL method, we get: 9x^2 - 39x - 6x + 26. Each term here represents the result of one of the four multiplications we performed. It is crucial to perform these multiplications accurately, paying close attention to the signs of each term. After this expansion, the next step is to simplify the expression by combining like terms. This will give us the most concise form of the expression. In the subsequent section, we will walk through the simplification process to arrive at our final answer.

Combining Like Terms for Simplification

After expanding the expression (3x - 2)(3x - 13) using the FOIL method, we arrived at 9x^2 - 39x - 6x + 26. The next step is to simplify this expression by combining like terms. Like terms are terms that have the same variable raised to the same power. In our expression, -39x and -6x are like terms because they both contain the variable x raised to the power of 1. The term 9x^2 has x raised to the power of 2 and 26 is a constant term, so they are not like terms with -39x and -6x. To combine like terms, we simply add or subtract their coefficients (the numbers in front of the variable). In this case, we need to combine -39x and -6x. Adding these terms together, we get -39x - 6x = -45x. Now, we substitute this back into our expression. The simplified expression becomes 9x^2 - 45x + 26. This is the simplest form of the original expression because there are no more like terms to combine. The process of combining like terms is essential for simplifying algebraic expressions and making them easier to work with. It reduces the complexity of the expression and allows for easier manipulation in further mathematical operations. In the next section, we will summarize the entire process and present the final simplified expression.

Final Simplified Expression

Having gone through the steps of expanding and simplifying the expression (3x - 2)(3x - 13), we can now present the final result. We began by understanding the distributive property and applying the FOIL method to expand the expression. This gave us 9x^2 - 39x - 6x + 26. Next, we identified and combined the like terms, specifically -39x and -6x, which resulted in -45x. Thus, the final simplified expression is: 9x^2 - 45x + 26. This expression is a quadratic trinomial, a common form in algebra. It is crucial to understand that this simplified form is equivalent to the original expression, but it is more concise and easier to work with in many contexts. For instance, if you needed to solve a quadratic equation involving this expression, having it in the simplified form makes the process much more straightforward. This exercise demonstrates the importance of mastering algebraic manipulation techniques. The ability to expand and simplify expressions is a fundamental skill that underpins more advanced mathematical concepts. By following the steps outlined in this guide, you can confidently tackle similar problems and enhance your algebraic proficiency. The journey from the initial expression to the simplified form highlights the power of algebraic techniques in making complex expressions manageable and understandable.

Conclusion

In conclusion, we have successfully expanded and simplified the expression (3x - 2)(3x - 13) to its final form: 9x^2 - 45x + 26. This process involved understanding and applying the distributive property, utilizing the FOIL method for expansion, and combining like terms for simplification. Each step is crucial in transforming the initial expression into a more concise and usable form. Mastering these algebraic techniques is essential for anyone studying mathematics, as they form the foundation for more advanced concepts. The ability to manipulate expressions effectively is not only beneficial in academic settings but also in practical applications where mathematical problem-solving is required. By breaking down the problem into manageable steps, we have demonstrated how complex algebraic tasks can be simplified and understood. This approach reinforces the importance of methodical problem-solving and the power of fundamental mathematical principles. The final simplified expression is not just an answer; it is a testament to the effectiveness of algebraic manipulation techniques in making complex problems more accessible. This skill will undoubtedly prove valuable in future mathematical endeavors. Remember, practice is key to mastering these concepts, so continue to apply these techniques to various expressions to solidify your understanding and proficiency.