Product Of (2x-5)(2x+5) Explained A Step By Step Guide

The question at hand is a classic example of a mathematical problem that tests our understanding of algebraic identities, specifically the difference of squares. The expression (2x-5)(2x+5) may seem complex at first glance, but it can be simplified elegantly using a fundamental algebraic principle. This article aims to dissect this problem, provide a step-by-step solution, and delve into the broader implications of such algebraic manipulations. We'll explore the underlying concept of the difference of squares, its applications in various mathematical contexts, and how mastering these techniques can significantly enhance your problem-solving abilities in algebra and beyond.

Understanding the question is crucial before attempting a solution. The problem asks us to find the product of two binomials: (2x-5) and (2x+5). In mathematical terms, the "product" refers to the result obtained when two or more numbers or expressions are multiplied together. In this case, we need to multiply the binomial (2x-5) by the binomial (2x+5). This involves applying the distributive property of multiplication, which states that each term in the first binomial must be multiplied by each term in the second binomial. However, recognizing the specific form of these binomials allows us to take a shortcut using the difference of squares identity. The binomials (2x-5) and (2x+5) are conjugates of each other, meaning they have the same terms but opposite signs. This particular structure hints at the applicability of the difference of squares formula, which simplifies the multiplication process significantly. By understanding this fundamental aspect of the problem, we can approach the solution with greater confidence and efficiency. The ability to recognize patterns and apply relevant algebraic identities is a cornerstone of mathematical proficiency, and this problem serves as an excellent opportunity to hone these skills. Let's now delve into the step-by-step solution and unveil the product of these two binomials.

To determine the product of (2x-5)(2x+5), we can employ the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first binomial is multiplied by each term in the second binomial. Let's break down the process:

1. First: Multiply the first terms of each binomial: 2x * 2x = 4x²
2. Outer: Multiply the outer terms of the binomials: 2x * 5 = 10x
3. Inner: Multiply the inner terms of the binomials: -5 * 2x = -10x
4. Last: Multiply the last terms of each binomial: -5 * 5 = -25

Now, we combine these results:

4x² + 10x - 10x - 25

Notice that the middle terms, 10x and -10x, are additive inverses and cancel each other out:

4x² - 25

Thus, the product of (2x-5)(2x+5), obtained through step-by-step expansion, is 4x² - 25. This method provides a straightforward approach to multiplying binomials and reinforces the application of the distributive property. However, recognizing the specific pattern of these binomials allows us to use a more efficient method, which we will explore next.

Recognizing the difference of squares pattern is a crucial skill in algebra. The binomials (2x-5) and (2x+5) fit this pattern perfectly. The difference of squares identity states that (a - b)(a + b) = a² - b². This identity provides a shortcut for multiplying conjugate pairs, which are binomials that have the same terms but opposite signs.

In our case, we can identify 'a' as 2x and 'b' as 5. Applying the difference of squares identity, we have:

(2x - 5)(2x + 5) = (2x)² - (5)²

Now, we simply square each term:

(2x)² = 4x² (5)² = 25

Therefore, the product is:

4x² - 25

This method demonstrates the elegance and efficiency of using algebraic identities. By recognizing the difference of squares pattern, we bypassed the need for the step-by-step expansion and arrived at the answer directly. This approach not only saves time but also highlights the importance of pattern recognition in mathematical problem-solving. Mastering such identities enhances your algebraic toolkit and allows you to tackle problems with greater speed and accuracy. Now, let's delve into the answer choices provided and identify the correct one.

Having calculated the product of (2x-5)(2x+5) using both the distributive property and the difference of squares identity, we have arrived at the solution: 4x² - 25. Now, we must compare this result with the provided answer choices to identify the correct one.

The answer choices are:

• A. 4x² - 25
• B. 4x² + 20x - 10
• C. 4x² + 20x - 25
• D. 4x² - 10

By direct comparison, it is clear that answer choice A. 4x² - 25 matches our calculated product. The other options contain terms that are not present in the correct expansion or have incorrect signs. For instance, options B and C include a 20x term, which arises from an incorrect application of the distributive property. Option D, on the other hand, only considers the squares of the terms but misses the crucial negative sign resulting from the difference of squares identity.

Therefore, the correct answer is definitively A. 4x² - 25. This exercise reinforces the importance of careful calculation and comparison when selecting the correct answer in mathematical problems. Now, let's consider why understanding this type of problem is crucial in the broader context of mathematics.

Understanding the product of expressions like (2x-5)(2x+5) is not just about solving a single problem; it's about grasping fundamental algebraic concepts that are crucial for more advanced mathematics. The ability to expand and simplify algebraic expressions, particularly those involving binomials and the difference of squares, forms the bedrock of many mathematical topics.

Firstly, mastering these skills is essential for solving equations. Many equations, especially quadratic equations, can be solved by factoring or by using the quadratic formula. Recognizing patterns like the difference of squares can significantly simplify the factoring process, making it easier to find the solutions to the equation. For example, if you encounter an equation like 4x² - 25 = 0, recognizing the left side as a difference of squares allows you to quickly factor it into (2x-5)(2x+5) = 0, leading to the solutions x = 5/2 and x = -5/2.

Secondly, these concepts are vital in calculus. Calculus often involves manipulating complex algebraic expressions, including those with binomials and polynomials. Understanding how to expand, simplify, and factor these expressions is essential for differentiation and integration. For instance, you might need to simplify an expression involving a difference of squares before applying a derivative rule or an integration technique.

Thirdly, the difference of squares and similar algebraic identities are used extensively in various fields of science and engineering. From physics to computer science, these principles find applications in areas such as signal processing, circuit analysis, and cryptography. A strong foundation in algebra allows students to approach these fields with greater confidence and understanding.

Finally, solving problems like the one we addressed cultivates critical thinking and problem-solving skills. It teaches you to recognize patterns, apply appropriate strategies, and break down complex problems into smaller, manageable steps. These skills are valuable not only in mathematics but also in various aspects of life. By mastering these fundamental concepts, you build a strong mathematical foundation that will serve you well in your academic and professional pursuits. To solidify your understanding, let's explore some similar problems and practice applying these concepts.

To further solidify your understanding of the difference of squares and its applications, let's explore some similar problems that you can practice. These exercises will help you develop your pattern recognition skills and your ability to apply the appropriate algebraic identities.

Problem 1: Find the product of (3x - 2)(3x + 2).

This problem is a direct application of the difference of squares identity. Identify 'a' and 'b' in this case, and apply the formula (a - b)(a + b) = a² - b². Compare your result with the expanded form obtained using the distributive property.

Problem 2: Simplify the expression (5y + 4)(5y - 4).

Again, this is a classic difference of squares scenario. Notice that the terms are the same, but the signs are opposite. Apply the identity to simplify the expression efficiently.

Problem 3: What is the result of expanding (4a - 7)(4a + 7)?

This problem reinforces the concept and provides an opportunity to practice the difference of squares identity. Make sure to square both terms correctly and pay attention to the signs.

Problem 4: Expand and simplify (x² - 3)(x² + 3).

This problem introduces a slight variation by using x² as one of the terms. However, the difference of squares identity still applies. Be careful when squaring x².

Problem 5: Calculate the product of (2p - 9)(2p + 9).

This problem is another straightforward application of the difference of squares. It provides an opportunity to practice the identity and build confidence.

By working through these problems, you will gain a deeper understanding of the difference of squares identity and its applications. Remember to focus on recognizing the pattern, applying the identity correctly, and simplifying the resulting expression. Practice is key to mastering these algebraic techniques and building a strong foundation for more advanced mathematical concepts. As a final thought, let's summarize the key takeaways from this discussion.

In conclusion, the product of (2x-5)(2x+5) is 4x² - 25. This problem serves as an excellent illustration of the difference of squares identity, a fundamental concept in algebra. Throughout this article, we have explored the problem from various angles, emphasizing the importance of understanding algebraic identities and their applications.

We began by understanding the problem statement and recognizing the specific form of the binomials. We then solved the problem using two methods: step-by-step expansion using the distributive property and applying the difference of squares identity. The latter method proved to be more efficient, highlighting the power of pattern recognition in mathematics. We also identified the correct answer choice from the given options, reinforcing the need for careful comparison and calculation.

Furthermore, we discussed the broader implications of understanding this type of problem. Mastering these algebraic skills is crucial for solving equations, simplifying expressions in calculus, and applying mathematical principles in various fields of science and engineering. It also cultivates critical thinking and problem-solving skills that are valuable in all aspects of life.

By practicing similar problems, you can solidify your understanding of the difference of squares and its applications. Remember to focus on recognizing the pattern, applying the identity correctly, and simplifying the resulting expressions.

The key takeaways from this discussion are:

• The difference of squares identity states that (a - b)(a + b) = a² - b².
• Recognizing patterns and applying appropriate algebraic identities can significantly simplify problem-solving.
• Mastering these skills is crucial for more advanced mathematical topics and various real-world applications.
• Practice is essential for building a strong foundation in algebra.

By keeping these points in mind and continuing to practice, you will develop a deeper understanding of algebra and its applications, empowering you to tackle a wide range of mathematical challenges with confidence.