Expanding (2x + 6)^2 A Step-by-Step Solution

In the realm of mathematics, polynomial expressions stand as fundamental building blocks. They form the basis for numerous equations, functions, and mathematical models that describe the world around us. Polynomials are algebraic expressions comprising variables and coefficients, connected by mathematical operations such as addition, subtraction, and multiplication, with non-negative integer exponents. Mastering the art of manipulating and simplifying polynomials is crucial for success in algebra, calculus, and beyond. One key skill within this domain is the expansion of squared binomials, a process that unlocks the hidden structure within these expressions and paves the way for further analysis and problem-solving. In this article, we will embark on a journey to unravel the intricacies of expanding the squared binomial (2x + 6)², exploring the underlying principles and techniques that empower us to simplify such expressions with confidence and precision. This comprehensive guide will not only provide a step-by-step solution to the problem at hand but also delve into the broader context of polynomial expansion, equipping you with the knowledge and skills to tackle a wide range of similar challenges. Whether you are a student seeking to solidify your understanding of algebra or a seasoned mathematician looking for a refresher, this article will serve as a valuable resource in your quest for mathematical mastery. So, let us embark on this mathematical adventure together, unlocking the secrets of polynomial expansion and gaining a deeper appreciation for the elegance and power of algebraic expressions.

At the heart of our exploration lies the expression (2x + 6)², a squared binomial that holds a wealth of mathematical information within its compact form. To fully understand and appreciate the expression, we must first dissect its components and grasp the operations involved. The expression is composed of two terms, 2x and 6, which are connected by the addition operation. These two terms together form a binomial, a polynomial with two terms. The superscript 2 indicates that the entire binomial (2x + 6) is raised to the power of 2, which signifies that it is multiplied by itself. In essence, (2x + 6)² is shorthand for (2x + 6) * (2x + 6). Now that we have deciphered the structure of the expression, the question becomes: how do we expand this squared binomial? Expansion, in this context, refers to the process of multiplying the binomial by itself and simplifying the resulting expression. This involves applying the distributive property, a fundamental principle in algebra that governs how multiplication interacts with addition and subtraction. The distributive property, in its simplest form, states that a(b + c) = ab + ac. This principle extends to the multiplication of binomials, where each term in the first binomial must be multiplied by each term in the second binomial. Expanding (2x + 6)² requires us to carefully apply the distributive property, ensuring that every term is accounted for and multiplied correctly. This process will unveil the underlying structure of the expression, transforming it from a compact squared binomial into a more expanded polynomial form. This expanded form will then reveal the coefficients and terms that make up the expression, providing us with a deeper understanding of its behavior and properties. In the following sections, we will delve into the step-by-step process of expanding (2x + 6)², illustrating the application of the distributive property and the simplification of the resulting expression. By the end of this journey, you will have a firm grasp of the techniques involved and the confidence to tackle similar challenges with ease.

The expansion of (2x + 6)² is a systematic process that involves the careful application of the distributive property and the combination of like terms. Let us embark on this step-by-step journey, unraveling the expression and revealing its expanded form. Step 1: Rewrite the expression. The first step is to rewrite the squared binomial as a product of two identical binomials. This clarifies the operation that needs to be performed and sets the stage for applying the distributive property. (2x + 6)² = (2x + 6)(2x + 6) Step 2: Apply the distributive property (often referred to as the FOIL method). The distributive property, in this context, dictates that each term in the first binomial must be multiplied by each term in the second binomial. A helpful mnemonic for this process is FOIL, which stands for First, Outer, Inner, Last. This acronym guides us through the multiplication steps: Multiply the First terms: 2x * 2x = 4x² Multiply the Outer terms: 2x * 6 = 12x Multiply the Inner terms: 6 * 2x = 12x Multiply the Last terms: 6 * 6 = 36 Combining these products, we get: (2x + 6)(2x + 6) = 4x² + 12x + 12x + 36 Step 3: Combine like terms. 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 this case, we have two terms with the variable x raised to the power of 1: 12x and 12x. Combining these terms, we get: 12x + 12x = 24x Now, we can rewrite the expression with the combined like terms: 4x² + 12x + 12x + 36 = 4x² + 24x + 36 Step 4: Write the final expanded form. The final step is to write the expression in its simplified, expanded form. This involves arranging the terms in descending order of their exponents, a standard convention in polynomial representation. The expanded form of (2x + 6)² is: 4x² + 24x + 36 And there you have it! We have successfully expanded the squared binomial (2x + 6)² by meticulously applying the distributive property and combining like terms. The expanded form, 4x² + 24x + 36, reveals the underlying structure of the expression and provides a foundation for further analysis and manipulation.

Now that we have meticulously expanded the expression (2x + 6)², let us turn our attention to the multiple-choice options provided and identify the correct answer. This involves comparing our expanded form with each option and determining which one matches perfectly. Our expanded form, as we derived in the previous section, is: 4x² + 24x + 36 Let us now examine each of the given options: A. 4x² + 36 This option is missing the middle term, 24x, which arises from the multiplication of the outer and inner terms in the binomial expansion. Therefore, option A is incorrect. B. 4x² - 24x + 36 This option has the correct first and last terms, but the middle term has the wrong sign. The middle term should be positive 24x, not negative 24x. Therefore, option B is incorrect. C. 4x² + 12x + 36 This option has the correct first and last terms, but the middle term has the wrong coefficient. The middle term should be 24x, not 12x. Therefore, option C is incorrect. D. 4x² + 24x + 36 This option perfectly matches our expanded form. It has the correct first term (4x²), the correct middle term (24x), and the correct last term (36). Therefore, option D is the correct answer. Detailed Explanation of Option D: Option D, 4x² + 24x + 36, is the correct answer because it accurately represents the expanded form of (2x + 6)². Each term in this option corresponds to the terms we derived during the expansion process: The term 4x² results from multiplying the first terms of the binomials: (2x)(2x) = 4x² The term 24x results from combining the products of the outer and inner terms: (2x)(6) + (6)(2x) = 12x + 12x = 24x The term 36 results from multiplying the last terms of the binomials: (6)(6) = 36 Therefore, option D encapsulates the complete and accurate expansion of the given expression. In conclusion, by carefully expanding the binomial and comparing our result with the provided options, we have confidently identified option D as the correct answer. This process underscores the importance of meticulous expansion and the accurate identification of like terms in polynomial simplification.

While we have successfully expanded (2x + 6)² through the application of the distributive property, there exists a more general formula that streamlines this process for any squared binomial. This formula, known as the square of a binomial formula, provides a shortcut for expanding expressions of the form (a + b)². The square of a binomial formula is expressed as: (a + b)² = a² + 2ab + b² This formula states that the square of a binomial is equal to the sum of the square of the first term (a²), twice the product of the two terms (2ab), and the square of the second term (b²). Understanding and applying this formula can significantly expedite the expansion of squared binomials, saving time and effort in algebraic manipulations. Let us now apply this formula to our original problem, (2x + 6)², to demonstrate its effectiveness and verify our previous result. In this case, a = 2x and b = 6. Substituting these values into the formula, we get: (2x + 6)² = (2x)² + 2(2x)(6) + (6)² Now, let us simplify each term: (2x)² = 4x² 2(2x)(6) = 24x (6)² = 36 Combining these terms, we arrive at the same expanded form we obtained through the distributive property: (2x + 6)² = 4x² + 24x + 36 The square of a binomial formula not only provides a shortcut for expansion but also offers valuable insights into the structure of squared binomials. It highlights the relationship between the terms of the binomial and the terms of its expansion, revealing the underlying pattern that governs this algebraic operation. Furthermore, the square of a binomial formula can be extended to cases where the binomial involves subtraction, such as (a - b)². In this case, the formula becomes: (a - b)² = a² - 2ab + b² The only difference is the sign of the middle term, which becomes negative due to the subtraction in the binomial. Mastering the square of a binomial formula is an essential skill in algebra, enabling us to efficiently expand squared binomials and gain a deeper understanding of polynomial expressions. It is a tool that empowers us to tackle a wide range of algebraic problems with confidence and precision.

To solidify your understanding of expanding squared binomials, it is crucial to engage in practice. Working through various examples will not only reinforce the concepts we have discussed but also help you develop fluency and confidence in applying the techniques. Here are some practice problems that will allow you to hone your skills: 1. Expand (3x + 4)² 2. Expand (5x - 2)² 3. Expand (x + 7)² 4. Expand (4x - 1)² 5. Expand (2x + 9)² For each of these problems, we encourage you to follow the steps we have outlined in this article: Rewrite the expression as a product of two binomials. Apply the distributive property (FOIL method). Combine like terms. Write the final expanded form. Alternatively, you can apply the square of a binomial formula directly: (a + b)² = a² + 2ab + b² or (a - b)² = a² - 2ab + b² By working through these problems, you will gain a deeper appreciation for the mechanics of expanding squared binomials and the power of the square of a binomial formula. You will also develop the ability to recognize patterns and apply the appropriate techniques to different types of expressions. Remember, practice is key to mastering any mathematical skill. The more you practice, the more comfortable and confident you will become in expanding squared binomials and tackling other algebraic challenges. In addition to these practice problems, you can also find numerous online resources and textbooks that offer additional exercises and examples. Seek out opportunities to challenge yourself and expand your knowledge. With consistent effort and practice, you will become a proficient manipulator of polynomial expressions, unlocking new avenues for mathematical exploration and problem-solving. So, grab a pencil and paper, and let's dive into these practice problems. The journey to mathematical mastery is an ongoing one, and each problem you solve brings you one step closer to your goal.

In this comprehensive exploration, we have delved into the intricacies of expanding squared binomials, unlocking the secrets hidden within these algebraic expressions. We began by understanding the problem, dissecting the structure of (2x + 6)² and recognizing the need for expansion. We then embarked on a step-by-step solution, meticulously applying the distributive property and combining like terms to arrive at the expanded form: 4x² + 24x + 36. We further solidified our understanding by identifying the correct answer from a set of multiple-choice options, providing a detailed explanation for why option D was the perfect match. To generalize our knowledge, we introduced the square of a binomial formula, a powerful tool that streamlines the expansion process for any squared binomial. This formula, (a + b)² = a² + 2ab + b², not only provides a shortcut but also reveals the underlying structure and patterns within squared binomials. We also explored the variation of the formula for binomials involving subtraction, (a - b)² = a² - 2ab + b², highlighting the subtle difference in the sign of the middle term. To reinforce our learning, we presented a set of practice problems, encouraging you to apply the techniques and formulas we have discussed. Practice, as we emphasized, is the cornerstone of mathematical mastery, and consistent effort will lead to fluency and confidence in expanding squared binomials. By diligently working through these problems, you will develop the ability to recognize patterns, apply the appropriate techniques, and tackle a wide range of algebraic challenges. The journey to mastering polynomial expansion is an ongoing one, but with the knowledge and skills you have gained from this article, you are well-equipped to continue your exploration. Remember, mathematics is not just about memorizing formulas and procedures; it is about understanding the underlying concepts and applying them creatively to solve problems. So, embrace the challenge, continue to practice, and never stop exploring the fascinating world of algebra. The art of polynomial expansion is a valuable tool in your mathematical arsenal, and with continued effort, you will wield it with precision and skill.