Evaluate 3(x-4) + 2x - X² For X=5 A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of evaluating algebraic expressions. Specifically, we will focus on the expression 3(x-4) + 2x - x² and determine its value when x equals 5. This type of problem is fundamental in algebra and serves as a building block for more complex mathematical concepts. Understanding how to substitute values into expressions and simplify them is crucial for success in mathematics. We will break down the steps involved in solving this problem, ensuring clarity and a thorough understanding of the underlying principles. This includes the order of operations, the distributive property, and the combination of like terms. By the end of this discussion, you will be equipped with the knowledge and skills to tackle similar evaluation problems with confidence. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide a comprehensive approach to solving the given expression.

Understanding the Problem

Before we begin to solve the problem, it's essential to understand what the question is asking. We are given an algebraic expression, 3(x-4) + 2x - x², and we are asked to find its value when x is equal to 5. This means we need to substitute 5 for every instance of x in the expression and then simplify the resulting numerical expression using the correct order of operations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Ignoring this order can lead to incorrect results. In our expression, we have parentheses, multiplication, addition, subtraction, and an exponent, so we must carefully follow PEMDAS to arrive at the correct answer. This step-by-step approach ensures accuracy and clarity in our solution. Furthermore, it's important to pay attention to the signs of the terms and to combine like terms correctly. Attention to detail is key when evaluating algebraic expressions, as a small mistake can lead to a wrong final answer. By understanding the problem thoroughly, we set the stage for a successful solution.

Step-by-Step Solution

To evaluate the expression 3(x-4) + 2x - x² for x = 5, we will follow these steps meticulously:

  1. Substitution: Replace every instance of x with 5. This gives us: 3(5-4) + 2(5) - (5)².
  2. Parentheses: Evaluate the expression inside the parentheses first. 5 - 4 = 1, so our expression becomes: 3(1) + 2(5) - (5)².
  3. Exponents: Next, we evaluate the exponent. 5² = 25, so the expression is now: 3(1) + 2(5) - 25.
  4. Multiplication: Perform the multiplications from left to right. 3(1) = 3 and 2(5) = 10. The expression becomes: 3 + 10 - 25.
  5. Addition and Subtraction: Finally, perform the addition and subtraction from left to right. 3 + 10 = 13, and then 13 - 25 = -12. Therefore, the final value of the expression is -12.

Each of these steps is crucial in arriving at the correct answer. By breaking down the problem into smaller, manageable parts, we can minimize the chance of errors and ensure a clear understanding of the process. This step-by-step approach not only helps in solving this particular problem but also provides a framework for tackling other algebraic evaluations. Remember to always double-check your work and ensure that you have followed the order of operations correctly. Practice makes perfect, so the more you work through these types of problems, the more confident you will become in your algebraic skills.

Detailed Calculation

Let's walk through the calculation in detail to ensure every step is clear:

  1. Substitution:

    • We begin by substituting x = 5 into the expression 3(x-4) + 2x - x². This yields: 3(5-4) + 2(5) - (5)².

  2. Parentheses:

    • We evaluate the expression within the parentheses: 5 - 4 = 1. So, the expression becomes: 3(1) + 2(5) - (5)².

  3. Exponents:

    • Next, we address the exponent: (5)² which means 5 * 5 = 25. The expression now reads: 3(1) + 2(5) - 25.

  4. Multiplication:

    • We perform the multiplications from left to right. First, 3(1) = 3. The expression becomes: 3 + 2(5) - 25.
    • Then, we multiply 2 by 5, which gives us 10. The expression is now: 3 + 10 - 25.

  5. Addition and Subtraction:

    • Finally, we perform addition and subtraction from left to right. First, we add 3 and 10, resulting in 13. The expression simplifies to: 13 - 25.
    • Subtracting 25 from 13 gives us -12. Therefore, the final answer is -12.

This detailed breakdown highlights the importance of each step in the order of operations. By following this methodical approach, we can confidently evaluate complex expressions and minimize the risk of errors. The process of substitution, simplification, and calculation must be executed with precision to arrive at the correct result. Understanding each operation and its placement within the order of operations is key to mastering algebraic evaluations. With careful attention to detail and consistent practice, one can become proficient in handling such expressions.

Identifying the Correct Answer

After performing the step-by-step evaluation of the expression 3(x-4) + 2x - x² for x = 5, we arrived at the value of -12. Now, let's look at the given options and identify the correct answer:

A. -12 B. 0 C. -4 D. 40

Comparing our result (-12) with the options provided, it is clear that option A, -12, is the correct answer. This confirms that our step-by-step calculation was accurate and that we followed the correct order of operations. Identifying the correct answer is a crucial final step in problem-solving. It ensures that we have not only performed the calculations correctly but also understood the question and its context. In this case, our detailed calculation and careful adherence to PEMDAS led us to the accurate solution, which matches option A. Always double-check your work and compare your result with the given options to ensure you have selected the right answer. This practice will help you build confidence in your problem-solving skills and improve your accuracy in mathematical evaluations.

Common Mistakes to Avoid

When evaluating 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. Here are some common mistakes to watch out for:

  1. Incorrect Order of Operations: Failing to follow the correct order of operations (PEMDAS) is a frequent error. For example, performing addition before multiplication or subtraction before exponents can lead to incorrect results. Always remember to prioritize Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

  2. Sign Errors: Making mistakes with negative signs is another common issue. For instance, not properly distributing a negative sign across terms within parentheses or incorrectly squaring a negative number can lead to errors. Pay close attention to the signs of each term and ensure they are handled correctly.

  3. Incorrect Substitution: Substituting the value of the variable incorrectly can derail the entire calculation. Double-check that you have replaced every instance of the variable with the correct value.

  4. Arithmetic Errors: Simple arithmetic errors, such as miscalculating a multiplication or addition, can also lead to wrong answers. Take your time and double-check your calculations to minimize these types of mistakes.

  5. Forgetting to Combine Like Terms: In more complex expressions, failing to combine like terms can leave the expression unsimplified and potentially lead to errors later in the calculation. Make sure to combine all like terms before arriving at your final answer.

By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy and confidence in evaluating algebraic expressions. Practice and attention to detail are key to mastering these skills. Always review your work and look for potential errors before finalizing your answer.

Practice Problems

To solidify your understanding of evaluating algebraic expressions, let's look at some practice problems similar to the one we just solved. Working through these examples will help you apply the concepts we've discussed and build your problem-solving skills.

Practice Problem 1: Evaluate the expression 2(x + 3) - x² + 4x for x = 2.

Solution Steps:

  1. Substitute x = 2 into the expression: 2(2 + 3) - (2)² + 4(2).
  2. Evaluate the parentheses: 2 + 3 = 5, so the expression becomes: 2(5) - (2)² + 4(2).
  3. Evaluate the exponent: (2)² = 4, so the expression is now: 2(5) - 4 + 4(2).
  4. Perform the multiplications: 2(5) = 10 and 4(2) = 8. The expression becomes: 10 - 4 + 8.
  5. Perform the addition and subtraction from left to right: 10 - 4 = 6, and then 6 + 8 = 14. Therefore, the final value of the expression is 14.

Practice Problem 2: Evaluate the expression 5x - 3(x - 1) + x³ for x = -1.

Solution Steps:

  1. Substitute x = -1 into the expression: 5(-1) - 3(-1 - 1) + (-1)³.
  2. Evaluate the parentheses: -1 - 1 = -2, so the expression becomes: 5(-1) - 3(-2) + (-1)³.
  3. Evaluate the exponent: (-1)³ = -1, so the expression is now: 5(-1) - 3(-2) - 1.
  4. Perform the multiplications: 5(-1) = -5 and -3(-2) = 6. The expression becomes: -5 + 6 - 1.
  5. Perform the addition and subtraction from left to right: -5 + 6 = 1, and then 1 - 1 = 0. Therefore, the final value of the expression is 0.

By working through these practice problems, you can reinforce your understanding of the evaluation process and develop confidence in your ability to solve similar problems. Remember to follow the order of operations carefully and pay attention to signs and arithmetic details.

Conclusion

In this comprehensive guide, we have thoroughly explored the process of evaluating the algebraic expression 3(x-4) + 2x - x² for x = 5. We began by understanding the problem, emphasizing the importance of substituting the value correctly and adhering to the order of operations (PEMDAS). We then presented a step-by-step solution, meticulously breaking down each operation to ensure clarity and accuracy. The detailed calculation further illuminated the process, highlighting the significance of each step in arriving at the correct result. We identified the correct answer, -12, among the given options, and discussed common mistakes to avoid, such as sign errors, incorrect substitutions, and arithmetic mistakes. Finally, we provided practice problems to reinforce the concepts learned and build problem-solving skills.

By following this structured approach, you can confidently tackle similar algebraic evaluations. Remember, practice is key to mastering these skills. The more you work through these types of problems, the more proficient you will become in algebra. Understanding and applying the order of operations, paying attention to detail, and double-checking your work are crucial habits to develop. With consistent effort, you can build a strong foundation in algebra and excel in more advanced mathematical concepts.