Simplifying 3x(x-12x) + 3x² - 2(x-2)² Process And Product Statements
In this article, we will delve into the process of simplifying the algebraic expression 3x(x-12x) + 3x² - 2(x-2)². This involves applying the order of operations, combining like terms, and expanding squared expressions. We will break down each step, ensuring a clear understanding of the simplification process. Furthermore, we will identify the true statements about the simplification process and the resulting product. This comprehensive guide aims to equip you with the skills to tackle similar algebraic problems with confidence. Mastering algebraic simplification is crucial for success in mathematics, as it forms the foundation for solving equations, inequalities, and more complex mathematical concepts. Whether you're a student learning algebra or someone looking to refresh your skills, this guide provides a step-by-step approach to simplifying expressions effectively. We will also emphasize common pitfalls to avoid and strategies for checking your work, ensuring accuracy in your calculations. By the end of this article, you will be able to simplify the given expression and understand the underlying principles that govern algebraic manipulations.
Understanding the Expression
Before we begin simplifying, let's first understand the components of the expression 3x(x-12x) + 3x² - 2(x-2)². This expression consists of several terms, each involving the variable 'x' raised to different powers. We have terms with 'x', 'x²', and a squared binomial (x-2)². The presence of parentheses and exponents indicates the order in which we need to perform the operations. Specifically, we must address the parentheses first, followed by the exponents, then multiplication, and finally addition and subtraction. This order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for arriving at the correct simplified form. The expression also involves coefficients, which are the numerical values multiplying the variable terms. These coefficients play a key role when we combine like terms. Recognizing the structure of the expression is the first step towards successful simplification. By carefully examining each term and its relationship to the others, we can devise a strategy for simplifying the expression efficiently. This initial analysis helps prevent errors and ensures that we apply the correct operations in the appropriate sequence. In the following sections, we will break down the simplification process step-by-step, highlighting the rationale behind each operation.
Step-by-Step Simplification
Now, let's embark on the step-by-step simplification of the expression 3x(x-12x) + 3x² - 2(x-2)². The first step involves simplifying the expression within the first set of parentheses: (x - 12x). Combining these like terms, we get -11x. So, the expression now becomes 3x(-11x) + 3x² - 2(x-2)². Next, we perform the multiplication 3x(-11x), which yields -33x². The expression is now -33x² + 3x² - 2(x-2)². The next significant step is to address the squared binomial (x-2)². Squaring this expression means multiplying it by itself: (x-2)(x-2). Using the distributive property (or the FOIL method), we expand this as follows: x(x) + x(-2) - 2(x) - 2(-2), which simplifies to x² - 2x - 2x + 4, and further simplifies to x² - 4x + 4. Now, we substitute this back into the expression: -33x² + 3x² - 2(x² - 4x + 4). We now need to distribute the -2 across the terms inside the parentheses: -2(x²) - 2(-4x) - 2(4), which gives us -2x² + 8x - 8. The expression now reads -33x² + 3x² - 2x² + 8x - 8. Finally, we combine the like terms. We have three terms involving x²: -33x², 3x², and -2x². Combining these gives us -33x² + 3x² - 2x² = -32x². The remaining terms are 8x and -8. Thus, the simplified expression is -32x² + 8x - 8. This step-by-step process ensures that we handle each operation in the correct order, leading to the accurate simplified form.
Verifying the Solution
After simplifying an expression, it's crucial to verify the solution to ensure accuracy. One method for verification is to substitute a numerical value for 'x' in both the original expression and the simplified expression. If the results match, it increases our confidence in the correctness of the simplification. Let's choose a simple value, such as x = 1. Substituting x = 1 into the original expression 3x(x-12x) + 3x² - 2(x-2)², we get: 3(1)(1-12(1)) + 3(1)² - 2(1-2)². This simplifies to 3(1)(-11) + 3(1) - 2(-1)², which further simplifies to -33 + 3 - 2(1) = -33 + 3 - 2 = -32. Now, let's substitute x = 1 into the simplified expression -32x² + 8x - 8: -32(1)² + 8(1) - 8. This simplifies to -32(1) + 8 - 8 = -32 + 8 - 8 = -32. Since both the original expression and the simplified expression yield the same result (-32) when x = 1, this provides strong evidence that our simplification is correct. However, it's important to note that this single verification does not guarantee correctness for all values of 'x'. For a more rigorous check, one could substitute additional values or use a symbolic algebra system to verify the simplification. This verification step is essential to avoid carrying errors forward in subsequent calculations or problem-solving steps. By taking the time to verify our solution, we enhance the reliability of our work.
Identifying True Statements
Now that we have simplified the expression 3x(x-12x) + 3x² - 2(x-2)² to -32x² + 8x - 8, let's identify true statements about the process and the simplified product. One true statement is that the term -2(x-2)² is simplified by first squaring the expression (x-2). As demonstrated in our step-by-step simplification, we indeed expanded (x-2)² before multiplying by -2. Another true statement concerns the simplified product being a quadratic expression. A quadratic expression is a polynomial of degree 2, meaning the highest power of the variable is 2. In our simplified expression, -32x² + 8x - 8, the term -32x² has x raised to the power of 2, confirming that it is a quadratic expression. A further true statement could involve the coefficients of the simplified expression. For instance, the coefficient of the x² term is -32, the coefficient of the x term is 8, and the constant term is -8. Recognizing these coefficients is important for understanding the behavior of the quadratic expression. When identifying true statements, it's crucial to refer back to the steps taken during the simplification process and the final form of the expression. This ensures that the statements are accurate and supported by the work we have done. By carefully analyzing the process and the result, we can confidently identify the true statements that describe the simplification and the nature of the simplified product.
Common Mistakes to Avoid
When simplifying algebraic expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and improve your accuracy. One common mistake is incorrectly applying the distributive property. For example, when expanding -2(x² - 4x + 4), a student might forget to distribute the -2 to all three terms inside the parentheses, potentially leading to an incorrect result. Another frequent error occurs when combining like terms. Students may mistakenly combine terms that have different powers of the variable, such as adding x² and x terms together. It's crucial to remember that only terms with the same variable and exponent can be combined. A third common mistake involves errors in arithmetic, such as incorrect multiplication or addition of coefficients. Even a small arithmetic error can propagate through the entire simplification process, leading to a wrong answer. Another area prone to errors is handling negative signs. Forgetting to account for a negative sign during multiplication or distribution can significantly alter the result. For instance, -(-4x) should be simplified to +4x, but a student might overlook the double negative and incorrectly write -4x. Finally, students sometimes fail to follow the correct order of operations (PEMDAS). Skipping a step or performing operations in the wrong sequence can lead to incorrect simplification. By being mindful of these common mistakes and carefully checking each step of the process, you can minimize errors and achieve accurate results when simplifying algebraic expressions.
Conclusion
In conclusion, simplifying the expression 3x(x-12x) + 3x² - 2(x-2)² involves a series of steps, each requiring careful attention to detail. We began by understanding the expression's components and the order of operations. We then proceeded with the simplification process, addressing the parentheses, exponents, multiplication, and finally, combining like terms. The simplified form of the expression is -32x² + 8x - 8. We emphasized the importance of verifying the solution to ensure accuracy, demonstrating the substitution method as a valuable tool. Furthermore, we identified true statements about the simplification process and the resulting product, reinforcing our understanding of the expression's nature. We also highlighted common mistakes to avoid, such as misapplying the distributive property, incorrectly combining like terms, arithmetic errors, and neglecting the order of operations. By mastering the techniques and strategies discussed in this guide, you can confidently tackle similar algebraic simplification problems. Remember, practice is key to improving your skills and accuracy in mathematics. By consistently applying these methods and being mindful of potential errors, you can enhance your algebraic proficiency and achieve success in your mathematical endeavors. Simplifying expressions is a fundamental skill in algebra, and a solid understanding of this process will serve you well in more advanced mathematical topics.
