Simplifying Algebraic Expressions 3a - 7a² - 3a

In the realm of algebra, simplifying expressions is a fundamental skill. It involves manipulating an algebraic expression to make it more concise and easier to understand. Often, this involves combining like terms, which are terms that have the same variable raised to the same power. This article will delve into the process of simplifying the algebraic expression 3a - 7a² - 3a. We'll break down the steps, explain the underlying principles, and provide a comprehensive understanding of how to tackle similar problems. Mastering these techniques is crucial for success in more advanced mathematical concepts, such as solving equations, graphing functions, and calculus. The ability to simplify expressions not only makes calculations easier but also provides a clearer picture of the relationships between variables. By the end of this discussion, you'll have a solid grasp of how to simplify algebraic expressions and be well-equipped to tackle more complex problems.

The expression we're going to simplify is 3a - 7a² - 3a. Before we begin the simplification process, it's essential to understand the different components of this expression. The expression consists of three terms: 3a, -7a², and -3a. Each term is a combination of coefficients (the numerical part) and variables (the letters representing unknown values). In the term 3a, the coefficient is 3, and the variable is a. In the term -7a², the coefficient is -7, and the variable is a raised to the power of 2 (which means a squared). The last term, -3a, has a coefficient of -3 and a variable of a. The key to simplifying expressions is identifying like terms. Like terms are terms that have the same variable raised to the same power. For example, 3a and -3a are like terms because they both have the variable a raised to the power of 1 (even though it's not explicitly written, a is the same as a¹). However, -7a² is not a like term with 3a and -3a because it has the variable a raised to the power of 2. Understanding this distinction is crucial for combining terms correctly. We can only combine like terms through addition and subtraction. Terms with different variables or the same variable raised to different powers cannot be combined. This principle ensures that we maintain the integrity of the expression and arrive at an equivalent, but simpler, form. As we move forward, we'll apply this understanding to simplify the given expression systematically.

To effectively simplify the expression 3a - 7a² - 3a, the first crucial step is to pinpoint the like terms. As we discussed earlier, like terms are those that share the same variable raised to the same power. In our expression, we have three terms: 3a, -7a², and -3a. Let's analyze them closely to identify the ones that fit this criterion. The terms 3a and -3a both contain the variable a raised to the power of 1. Remember, when a variable has no explicit exponent, it is understood to be raised to the power of 1. This means that 3a is essentially 3a¹, and -3a is -3a¹. Since they have the same variable and the same power, 3a and -3a are indeed like terms. On the other hand, the term -7a² has the variable a raised to the power of 2. This makes it different from 3a and -3a. While it shares the same variable (a), the power is different. Therefore, -7a² is not a like term with the other two. The ability to accurately identify like terms is the cornerstone of simplifying algebraic expressions. Incorrectly grouping terms can lead to incorrect simplifications and ultimately affect the solution of any equation or problem the expression is part of. With the like terms identified as 3a and -3a, we can now move on to the next step, which is combining them.

Now that we've identified the like terms in the expression 3a - 7a² - 3a, the next step is to combine them. Combining like terms involves adding or subtracting their coefficients while keeping the variable and its exponent the same. In our case, the like terms are 3a and -3a. To combine these, we focus on their coefficients, which are 3 and -3, respectively. We perform the operation indicated in the expression, which is subtraction in this case. So, we have 3a - 3a. This is equivalent to (3 - 3)a. Performing the subtraction, we get 0. Therefore, 3a - 3a = 0a. Any term with a coefficient of 0 is equal to 0, so 0a simplifies to 0. This means that the terms 3a and -3a effectively cancel each other out in the expression. The remaining term in our original expression is -7a². Since there are no other terms with a², we cannot combine it with anything else. It remains as it is. This highlights an important principle in simplifying expressions: only like terms can be combined. Terms that have different variables or the same variable raised to different powers must be kept separate. By combining the like terms 3a and -3a, we've made significant progress in simplifying the expression. We've reduced two terms to a single value, 0. This makes the expression more concise and easier to work with. The next step is to write the simplified expression, taking into account the combined terms and any remaining terms.

After combining the like terms in the expression 3a - 7a² - 3a, we arrived at the result 0. However, we still have the term -7a² which was not a like term with any other in the original expression. To write the simplified expression, we need to consider both the result of combining the like terms and any remaining terms that couldn't be combined. In our case, we have the combined result of 3a - 3a which is 0, and the term -7a². When we write the simplified expression, we include all the terms, even if some of them are zero. So, we can write the simplified expression as 0 - 7a². However, adding or subtracting 0 from any expression doesn't change its value. Therefore, we can further simplify the expression by removing the 0. This gives us the final simplified expression: -7a². This expression is much simpler than the original expression 3a - 7a² - 3a. We've reduced three terms to just one term, making it easier to understand and work with. The simplified expression -7a² represents the same mathematical value as the original expression, but in a more concise form. This is the essence of simplifying algebraic expressions: to find an equivalent expression that is easier to handle. In this case, we've successfully simplified the expression by identifying and combining like terms and then writing the result in its simplest form. This process is a fundamental skill in algebra and is used extensively in solving equations, graphing functions, and other mathematical operations.

Therefore, after meticulously identifying and combining like terms, the simplified form of the expression 3a - 7a² - 3a is -7a². This final form is significantly more concise than the original expression, making it easier to understand and manipulate in further mathematical operations. The process of simplifying algebraic expressions is not merely about reducing the number of terms; it's about revealing the underlying structure and relationships within the expression. In this case, the simplified form –7a² clearly shows that the expression is a quadratic term, where the variable a is raised to the power of 2, and the coefficient is -7. This clarity can be invaluable when solving equations, graphing functions, or performing other algebraic manipulations. By simplifying expressions, we reduce the complexity and make it easier to see the essential components and their interactions. This, in turn, reduces the likelihood of errors and allows for a more efficient and accurate problem-solving process. The ability to simplify expressions is a cornerstone of algebra and is a skill that will be used throughout your mathematical journey. Mastering this skill not only makes calculations easier but also provides a deeper understanding of mathematical concepts. This example demonstrates the power of simplification in making algebraic expressions more manageable and transparent. This skill is not just applicable to simple expressions like this one but also extends to more complex algebraic manipulations and is essential for success in higher-level mathematics. Therefore, understanding and practicing the steps involved in simplifying expressions is crucial for building a strong foundation in algebra.

In summary, we've successfully simplified the algebraic expression 3a - 7a² - 3a to its most concise form: -7a². This process involved several key steps, each of which is essential for accurate simplification. First, we understood the expression and its components, identifying the terms and their coefficients and variables. Then, we focused on identifying like terms, which are terms that have the same variable raised to the same power. In our expression, the like terms were 3a and -3a. The next crucial step was to combine these like terms. We did this by adding or subtracting their coefficients while keeping the variable and its exponent the same. In this case, 3a - 3a equals 0, effectively canceling each other out. Finally, we wrote the simplified expression, which consisted of the remaining term, -7a². This resulting expression, -7a², is the simplified form of the original expression. Simplifying algebraic expressions is a fundamental skill in mathematics. It makes expressions easier to understand, manipulate, and work with in various mathematical operations. By mastering this skill, you'll be well-equipped to tackle more complex algebraic problems and concepts. This example provides a clear and step-by-step guide to the simplification process, which can be applied to other similar expressions. The ability to simplify expressions not only improves your problem-solving skills but also enhances your understanding of the underlying mathematical principles. This skill is crucial for success in algebra and beyond, laying the foundation for more advanced mathematical topics. Remember to always look for like terms and combine them carefully, ensuring that you maintain the integrity of the expression throughout the simplification process.