Evaluating (a - Bc) + (4ab + 3bc) + 2(b^2 - Ac) For A=-5, B=3, And C=6

by ADMIN 71 views

Introduction

In this article, we will meticulously evaluate the algebraic expression (a - bc) + (4ab + 3bc) + 2(b^2 - ac) given specific values for the variables a, b, and c. This exercise not only reinforces fundamental algebraic principles but also highlights the importance of accurate substitution and order of operations. We are given that a = -5, b = 3, and c = 6. Our goal is to substitute these values into the expression and simplify it step by step to arrive at a final numerical answer. The process involves careful handling of negative signs, multiplication, and addition, making it a comprehensive review of basic arithmetic and algebraic manipulation. By breaking down the problem into manageable parts, we can ensure clarity and minimize the chance of errors. This detailed walkthrough aims to provide a clear understanding of the evaluation process, which is crucial for more complex algebraic problems. Algebraic expressions are the backbone of mathematics, and mastering the art of evaluating them is essential for progress in higher-level mathematics and various applications in science and engineering. This article will serve as a practical guide for students and anyone looking to refresh their algebraic skills. The step-by-step approach will help reinforce the rules of arithmetic and the order of operations, ensuring a solid foundation for future mathematical endeavors.

Step-by-Step Evaluation

1. The importance of carefully substituting variables: Substitute the given values of a, b, and c into the expression. The expression is (a - bc) + (4ab + 3bc) + 2(b^2 - ac), and the values are a = -5, b = 3, and c = 6. Replacing the variables with their respective values is the crucial first step in evaluating any algebraic expression. Accuracy in this step is paramount, as any error in substitution will propagate through the rest of the calculation, leading to an incorrect final answer. It is essential to pay close attention to the signs of the numbers, particularly when dealing with negative values. In this case, a is negative, which means its sign must be carefully managed throughout the calculation. The process of substitution is not merely a mechanical replacement of variables with numbers; it requires a clear understanding of the expression's structure and how each term interacts with others. For instance, terms involving multiplication, such as bc and 4ab, need to be handled differently from terms involving addition or subtraction. This initial step sets the stage for the rest of the evaluation, and a methodical approach here will make the subsequent steps easier and more accurate. Taking the time to double-check the substitution ensures that the expression is correctly set up for the following calculations. Careful substitution minimizes the risk of making mistakes and lays a strong foundation for the rest of the solution. The initial substitution is as follows:

( -5 - (3)(6) ) + ( 4(-5)(3) + 3(3)(6) ) + 2( (3)^2 - (-5)(6) )

2. Perform multiplications within parentheses: The next step is to perform the multiplications within the parentheses. This involves calculating the products of the numbers within each set of parentheses, following the order of operations. Within the first set of parentheses, we have (3)(6), which equals 18. In the second set, we have 4(-5)(3), which equals -60, and 3(3)(6), which equals 54. In the third set, we have (3)^2, which equals 9, and (-5)(6), which equals -30. Accurate multiplication is critical at this stage, as it sets the values for the subsequent addition and subtraction operations. This step requires a firm understanding of the rules of multiplication, especially when dealing with negative numbers. The product of two negative numbers is positive, while the product of a positive and a negative number is negative. Maintaining the correct signs is essential for arriving at the correct final answer. By meticulously performing each multiplication, we reduce the expression to a simpler form, making it easier to handle in the following steps. This systematic approach is crucial for minimizing errors and ensuring a clear and accurate evaluation. Multiplication forms the core of many algebraic expressions, and mastering this step is essential for handling more complex problems. By breaking down the multiplications and solving them one at a time, we make the overall evaluation process more manageable and less prone to errors. The expression now becomes:

( -5 - 18 ) + ( -60 + 54 ) + 2( 9 - (-30) )

3. Continue simplifying inside the parentheses: Continuing with the order of operations, simplify the expressions inside each set of parentheses by performing the additions and subtractions. In the first set, -5 - 18 equals -23. In the second set, -60 + 54 equals -6. In the third set, we have 9 - (-30), which is equivalent to 9 + 30, resulting in 39. This step is crucial for condensing the expression into its simplest form before moving on to the next operations. Accurate arithmetic is paramount at this stage to avoid any carry-over errors. Simplifying within parentheses helps to isolate the values, making the subsequent steps of multiplication and addition more straightforward. The order of operations dictates that we resolve the innermost groupings first, and parentheses are among the highest priority. By carefully performing the additions and subtractions, we streamline the expression and prepare it for the final calculations. This process of simplification is not just about getting the right numbers; it's also about organizing the expression in a way that makes it easier to work with. The goal is to reduce the complexity and minimize the potential for mistakes. Each step in the simplification process brings us closer to the final answer, and maintaining accuracy throughout ensures a correct result. After simplifying within the parentheses, the expression looks like this:

(-23) + (-6) + 2(39)

4. Perform any remaining multiplication: Next, perform any remaining multiplication outside the parentheses. In this case, we have 2(39), which equals 78. This step involves multiplying the number outside the parentheses by the result of the expression inside. Accurate multiplication is critical to ensure that the final answer is correct. By performing this multiplication, we further simplify the expression and prepare it for the final addition. Multiplication distributes over addition and subtraction, so it's important to carry it out before any remaining addition or subtraction. This step is another key part of the order of operations, which helps to maintain consistency and accuracy in mathematical evaluations. By methodically performing the multiplication, we are one step closer to the final solution. The multiplication of 2 and 39 yields 78, which will be added to the other terms in the final step. This process highlights the importance of following the correct sequence of operations to ensure a reliable and accurate result. After performing the multiplication, the expression becomes:

-23 + (-6) + 78

5. Add the remaining terms: Finally, add the remaining terms to get the final answer. We have -23 + (-6) + 78. Adding -23 and -6 gives -29. Then, adding -29 to 78 gives 49. This final step involves combining all the simplified terms to arrive at the numerical result of the expression. Accurate addition is crucial in this stage, and it is important to pay attention to the signs of the numbers. Adding negative numbers requires careful handling, and it is often helpful to think of it as subtracting the absolute value of the number. The cumulative effect of all previous steps leads to this final calculation, making it a critical component of the overall evaluation process. The order in which we add the numbers does not affect the result, but it can be helpful to combine the negative numbers first to simplify the process. The addition of the terms represents the culmination of all the previous steps, and it provides the final, simplified value of the expression. This step reinforces the fundamental arithmetic principles of addition and subtraction, which are the building blocks of more complex mathematical operations. The final evaluation of the expression yields 49, representing the solution to the problem. After adding the remaining terms, the final result is:

-23 + (-6) + 78 = -29 + 78 = 49

Conclusion

In conclusion, by meticulously following the order of operations and carefully substituting the given values, we have successfully evaluated the expression (a - bc) + (4ab + 3bc) + 2(b^2 - ac) for a = -5, b = 3, and c = 6. The final result is 49. This exercise underscores the importance of accuracy and systematic problem-solving in algebra. Each step, from the initial substitution to the final addition, required careful attention to detail to avoid errors. Understanding and applying the correct order of operations is crucial for simplifying complex expressions and arriving at the correct answer. This evaluation process not only reinforces basic arithmetic skills but also builds a solid foundation for more advanced algebraic manipulations. By breaking down the problem into smaller, manageable steps, we can tackle even the most daunting expressions with confidence. The ability to accurately evaluate algebraic expressions is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and finance. This detailed walkthrough demonstrates how to approach such problems methodically, ensuring both accuracy and understanding. The final answer of 49 represents the culmination of a series of carefully executed steps, highlighting the power of methodical problem-solving in mathematics. Through practice and attention to detail, one can master the art of evaluating algebraic expressions and apply these skills to a wide range of mathematical challenges. This exercise serves as a valuable reminder of the importance of precision and the rewarding nature of mathematical problem-solving. Thus, the evaluated expression simplifies to 49, demonstrating the effectiveness of algebraic principles and careful calculation.