Simplifying Algebraic Expressions A Step By Step Guide To (2x - 9)(x + 6)

In mathematics, simplifying expressions is a fundamental skill. It involves rewriting an expression in a more concise and manageable form. This article delves into the process of simplifying the given algebraic expression: (2x - 9)(x + 6). We will explore the step-by-step method of expanding and combining like terms to arrive at the simplified form. This process is crucial not only for solving algebraic equations but also for understanding more advanced mathematical concepts. Mastering this technique will empower you to tackle a wide range of mathematical problems with confidence and precision. Throughout this discussion, we will emphasize the importance of accuracy and attention to detail, as even a small error can lead to an incorrect result. Our goal is to provide a clear and comprehensive guide that will help you understand and apply the principles of simplifying algebraic expressions effectively.

The distributive property is a cornerstone of algebraic manipulation and is essential for simplifying expressions like (2x - 9)(x + 6). This property allows us to multiply a single term by multiple terms within parentheses. To thoroughly understand the distributive property, let’s consider it in the context of our expression. The distributive property, in its simplest form, states that a(b + c) = ab + ac. However, when dealing with binomials (expressions with two terms), we extend this principle using the FOIL method (First, Outer, Inner, Last) or by distributing each term in the first binomial across each term in the second binomial. In the expression (2x - 9)(x + 6), we apply the distributive property by multiplying each term in the first binomial (2x and -9) by each term in the second binomial (x and 6). This systematic approach ensures that every term is accounted for, which is critical for achieving the correct simplified expression. Understanding the nuances of the distributive property is not just about memorizing a rule; it's about grasping the fundamental concept of how multiplication interacts with addition and subtraction in algebraic expressions. With a solid grasp of this property, simplifying more complex expressions becomes significantly more manageable.

To simplify the expression (2x - 9)(x + 6), we will meticulously apply the distributive property. This involves multiplying each term in the first binomial by each term in the second binomial. Here’s a detailed breakdown of each step:

1. Multiply the First terms: Multiply the first terms of each binomial, which are 2x and x. This gives us 2x * x = 2x². This is the first term of our simplified expression.
2. Multiply the Outer terms: Next, we multiply the outer terms, which are 2x and 6. This yields 2x * 6 = 12x. This result is an important component that we will combine with other terms later.
3. Multiply the Inner terms: Now, we multiply the inner terms, which are -9 and x. This gives us -9 * x = -9x. Pay close attention to the sign; the negative sign is crucial for the accuracy of the final result.
4. Multiply the Last terms: Finally, we multiply the last terms of each binomial, which are -9 and 6. This results in -9 * 6 = -54. Again, the negative sign is critical here.

After performing these multiplications, we have expanded the expression to 2x² + 12x - 9x - 54. The next step involves combining like terms to further simplify this expression. This methodical approach ensures that we do not miss any terms and that each multiplication is performed correctly, setting the stage for accurate simplification.

After expanding the expression (2x - 9)(x + 6), we arrive at 2x² + 12x - 9x - 54. The next crucial step in simplifying this expression is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are 12x and -9x. To combine them, we simply add or subtract their coefficients. In this case, we have 12x - 9x. Subtracting the coefficients, we get 12 - 9 = 3. Therefore, combining 12x and -9x results in 3x. The other terms in the expression, 2x² and -54, do not have like terms, so they remain unchanged. The term 2x² is a quadratic term, and -54 is a constant term. These terms cannot be combined with the 3x term, which is a linear term. Combining like terms is a critical step in simplifying algebraic expressions, as it reduces the expression to its most concise form. It’s important to ensure that you are only combining terms that have the same variable and exponent. This process not only simplifies the expression but also makes it easier to work with in further calculations or when solving equations.

After meticulously expanding and combining like terms, we arrive at the final simplified form of the expression (2x - 9)(x + 6). Starting with the expanded form 2x² + 12x - 9x - 54, we identified and combined the like terms 12x and -9x, which resulted in 3x. The terms 2x² and -54 remained unchanged as they did not have any like terms to combine with. Thus, the simplified expression is 2x² + 3x - 54. This quadratic expression is now in its simplest form, making it easier to understand and use in various mathematical contexts. The process of simplifying algebraic expressions is not just about arriving at a final answer; it’s about understanding the underlying principles of algebra and applying them systematically. Each step, from applying the distributive property to combining like terms, is a critical part of the process. A solid understanding of these steps is essential for success in algebra and beyond. The ability to simplify expressions accurately and efficiently is a valuable skill that will serve you well in more advanced mathematical studies.

In conclusion, simplifying the expression (2x - 9)(x + 6) involves a methodical application of algebraic principles. We began by understanding the distributive property, which is essential for expanding expressions involving binomials. We then meticulously applied this property, multiplying each term in the first binomial by each term in the second binomial. This process yielded the expanded form: 2x² + 12x - 9x - 54. The next critical step was combining like terms. By identifying and combining the terms 12x and -9x, we simplified the expression further. This led us to the final simplified form: 2x² + 3x - 54. This result is a quadratic expression in its simplest form, and it represents the most concise way to express the original product of binomials. The ability to simplify expressions like this is a fundamental skill in algebra and is crucial for solving equations, understanding functions, and tackling more complex mathematical problems. By mastering these techniques, you gain a deeper understanding of algebraic structures and enhance your problem-solving abilities. Remember, accuracy and attention to detail are key when simplifying expressions. Each step, from applying the distributive property to combining like terms, requires careful execution to ensure the final result is correct.

Therefore, the correct answer is A. 2x² + 3x - 54.