Solving (2x+3)(2x-3)-(25x+10) A Detailed Explanation

Jul 28, 2025 by ADMIN 53 views

Unveiling the Solution for (2x+3)(2x-3)-(25x+10) A Comprehensive Mathematical Discussion

Introduction to the Problem

In this mathematical discussion, we delve into the intricacies of simplifying and solving the expression (2x+3)(2x-3)-(25x+10). This problem encompasses several key algebraic concepts, including the difference of squares, distributive property, and combining like terms. A thorough exploration of these concepts will not only lead us to the solution but also enhance our understanding of algebraic manipulations. Before we proceed with the step-by-step solution, it's important to understand the fundamental principles that govern algebraic expressions. These include the order of operations (PEMDAS/BODMAS), the distributive property (a(b+c) = ab + ac), and the difference of squares identity (a² - b² = (a+b)(a-b)). Grasping these principles is crucial for solving not just this particular problem but also a wide range of algebraic challenges. When tackling such expressions, a systematic approach is key. We'll begin by expanding the product using the difference of squares identity, then distribute any constants, and finally combine like terms to arrive at the simplest form of the expression. This methodical approach minimizes the chances of errors and ensures a clear path to the solution. Moreover, understanding the underlying concepts allows us to tackle similar problems with confidence and efficiency. This problem serves as a great example to practice and reinforce algebraic skills, which are fundamental to more advanced mathematical concepts. As we dissect the problem, we will highlight common pitfalls and strategies to avoid them, making this discussion not only about finding the answer but also about developing a deeper understanding of algebraic problem-solving techniques. Our goal is to not just provide the solution but to empower you with the knowledge and skills to confidently tackle similar algebraic challenges in the future. So, let's embark on this mathematical journey and unravel the solution together, step by step.

Step-by-Step Solution

To effectively solve the expression (2x+3)(2x-3)-(25x+10), we'll employ a methodical approach, breaking it down into manageable steps. First and foremost, we recognize the structure of the expression and identify the key operations involved. This includes the multiplication of binomials and the subtraction of a linear expression. The initial part of the expression, (2x+3)(2x-3), presents a classic case of the difference of squares, a fundamental concept in algebra. The difference of squares identity states that a² - b² = (a+b)(a-b). Applying this identity here, where 'a' is 2x and 'b' is 3, we can simplify this product significantly. This simplification not only makes the expression easier to handle but also demonstrates a powerful algebraic shortcut. Instead of performing a full expansion using the distributive property (which would still lead to the same result), recognizing the difference of squares allows us to jump directly to the simplified form. After applying the difference of squares identity, we move on to the next part of the expression, which involves distributing the negative sign across the terms inside the parentheses. This step is crucial because neglecting the negative sign can lead to errors in the final result. Each term inside the parentheses must be multiplied by -1, effectively changing its sign. This distribution is a direct application of the distributive property, which is a cornerstone of algebraic manipulation. Once we've handled the distribution, the next step is to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, these include terms with 'x' and constant terms. Combining like terms simplifies the expression by grouping similar terms together, making it easier to read and understand. This process often involves adding or subtracting the coefficients of the like terms. Finally, after combining like terms, we arrive at the simplified form of the expression. This final form represents the solution to the problem and showcases the power of algebraic manipulation in reducing complex expressions to their simplest forms. Throughout this step-by-step solution, we emphasize the importance of accuracy and attention to detail. Each step builds upon the previous one, so errors early on can propagate through the entire solution. By carefully applying algebraic principles and following a methodical approach, we can confidently solve the expression and gain a deeper understanding of algebraic problem-solving.

Applying the Difference of Squares

To begin, we focus on the term (2x+3)(2x-3). As identified earlier, this is a classic example of the difference of squares. The difference of squares identity is a powerful tool in algebra, allowing us to quickly simplify expressions of this form. It's essential to recognize this pattern, as it significantly reduces the amount of computation required. Instead of employing the traditional method of expanding the product using the distributive property (often referred to as FOIL - First, Outer, Inner, Last), we can directly apply the identity. The difference of squares identity, a² - b² = (a+b)(a-b), provides a direct path to simplification. In our case, 'a' corresponds to 2x and 'b' corresponds to 3. Substituting these values into the identity, we can rewrite (2x+3)(2x-3) as (2x)² - (3)². This substitution is a crucial step in applying the identity correctly. It's important to ensure that the values are substituted in the correct places, paying attention to the order and signs. Once we have made the substitution, we need to simplify the squares. (2x)² is equivalent to 4x², and (3)² is 9. This simplification involves applying the exponent to both the coefficient and the variable, a fundamental concept in algebra. Therefore, the expression (2x)² - (3)² simplifies to 4x² - 9. This result is a direct consequence of applying the difference of squares identity and highlights its efficiency in simplifying such expressions. By recognizing the pattern and applying the identity, we have bypassed the need for a more lengthy expansion process. Now, we can substitute this simplified form back into the original expression, making it easier to handle the subsequent steps. This substitution demonstrates a key strategy in problem-solving: breaking down complex problems into smaller, more manageable parts. By simplifying one part of the expression, we have made the overall problem less daunting. Understanding and applying the difference of squares identity is not just about solving this particular problem; it's about developing a deeper understanding of algebraic patterns and techniques. This skill will prove invaluable in tackling more complex algebraic challenges in the future. So, let's continue our journey, armed with this simplified term, and unravel the remaining steps in the solution.

Distributing and Combining Like Terms

Having simplified the first part of the expression, (2x+3)(2x-3), to 4x² - 9, we now turn our attention to the remaining portion: -(25x+10). This step involves distributing the negative sign across the terms inside the parentheses. The negative sign in front of the parentheses acts as a multiplier of -1, which must be applied to each term inside. This is a direct application of the distributive property, a fundamental concept in algebra. The distributive property states that a(b+c) = ab + ac. In our case, 'a' is -1, 'b' is 25x, and 'c' is 10. Applying the distributive property, we multiply -1 by both 25x and 10. This results in -25x and -10, respectively. Therefore, -(25x+10) becomes -25x - 10. This step is crucial because neglecting the negative sign or failing to distribute it correctly can lead to significant errors in the final result. Now, we can rewrite the entire expression, substituting the simplified form and the distributed terms: 4x² - 9 - 25x - 10. With the expression in this form, the next step is to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, we have constant terms (-9 and -10) and a term with 'x' (-25x). The term 4x² is unique, as it is the only term with 'x' raised to the power of 2. To combine like terms, we simply add or subtract their coefficients. In this case, we combine the constant terms -9 and -10, which results in -19. The term -25x remains as is, as there are no other terms with 'x' to combine it with. Similarly, the term 4x² remains unchanged. After combining like terms, we can write the simplified expression in its final form. It is conventional to write the terms in descending order of their exponents, meaning the term with the highest exponent comes first. Therefore, the final simplified expression is 4x² - 25x - 19. This result represents the solution to the original problem and demonstrates the power of algebraic manipulation in reducing complex expressions to their simplest forms. Throughout this process, we have emphasized the importance of accuracy and attention to detail. Each step builds upon the previous one, so errors early on can propagate through the entire solution. By carefully applying algebraic principles and following a methodical approach, we have confidently solved the expression and gained a deeper understanding of algebraic problem-solving techniques.

Final Simplified Form

After meticulously applying the difference of squares identity, distributing the negative sign, and combining like terms, we have arrived at the final simplified form of the expression (2x+3)(2x-3)-(25x+10). This final form represents the culmination of our algebraic manipulations and provides a concise and understandable representation of the original expression. The journey to this simplified form involved several key steps, each requiring careful attention to detail and a solid understanding of algebraic principles. From recognizing the difference of squares pattern to correctly distributing the negative sign, each step played a crucial role in the overall solution. The final simplified form, 4x² - 25x - 19, is a quadratic expression. Quadratic expressions are a fundamental part of algebra and have numerous applications in mathematics and other fields. They are characterized by the presence of a term with a variable raised to the power of 2, in this case, 4x². The other terms in the expression, -25x and -19, are the linear term and the constant term, respectively. The coefficients of these terms, -25 and -19, are also important characteristics of the expression. The final simplified form not only provides a solution to the original problem but also opens up possibilities for further analysis and manipulation. For example, we could use this simplified form to find the roots of the expression (the values of 'x' that make the expression equal to zero), graph the expression, or use it in further algebraic calculations. The process of simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to reduce complex expressions to their most basic form, making them easier to understand and work with. This skill is essential for solving equations, graphing functions, and tackling more advanced mathematical concepts. Moreover, the final simplified form demonstrates the power of algebraic manipulation in revealing the underlying structure of an expression. By carefully applying algebraic principles, we can transform a seemingly complex expression into a simple and elegant form. This ability to manipulate and simplify expressions is a cornerstone of mathematical problem-solving. So, let's celebrate our achievement in arriving at the final simplified form. It represents not just the solution to a specific problem but also a testament to our understanding of algebraic principles and our ability to apply them effectively. This knowledge will serve us well in tackling future mathematical challenges.

Conclusion

In conclusion, our comprehensive discussion and step-by-step solution of the expression (2x+3)(2x-3)-(25x+10) has not only yielded the final simplified form, 4x² - 25x - 19, but has also illuminated several key algebraic concepts and techniques. This journey through algebraic manipulation has reinforced the importance of understanding and applying fundamental principles such as the difference of squares identity, the distributive property, and the process of combining like terms. Each step in the solution process was carefully dissected and explained, highlighting the significance of accuracy and attention to detail. From recognizing the difference of squares pattern to correctly distributing the negative sign and combining like terms, each operation played a crucial role in arriving at the final simplified form. The final simplified form, a quadratic expression, serves as a testament to the power of algebraic manipulation in reducing complex expressions to their simplest forms. This ability to simplify expressions is a fundamental skill in mathematics, enabling us to solve equations, graph functions, and tackle more advanced mathematical challenges. Throughout this discussion, we have emphasized not just the mechanics of solving the problem but also the underlying concepts and strategies involved. This deeper understanding will empower you to confidently tackle similar algebraic challenges in the future. We have also highlighted common pitfalls and strategies to avoid them, making this discussion not just about finding the answer but also about developing a robust problem-solving approach. The final simplified form is more than just a solution; it is a starting point for further exploration and analysis. It can be used to find the roots of the expression, graph the corresponding function, or serve as a building block for more complex algebraic calculations. This demonstrates the interconnectedness of mathematical concepts and the importance of mastering fundamental skills. In summary, this discussion has provided a thorough exploration of algebraic manipulation, showcasing the power and elegance of mathematical problem-solving. By understanding the concepts and techniques discussed, you are well-equipped to tackle a wide range of algebraic challenges and continue your journey in the world of mathematics. The final simplified form is not just an end, but a new beginning, a foundation upon which to build further mathematical knowledge and skills.