Evaluate X² + 5x + 1 When X = -2

Jul 12, 2025 by ADMIN 33 views

Evaluate the Expression x² + 5x + 1 when x = -2

In mathematics, evaluating expressions is a fundamental skill. This article provides a comprehensive, step-by-step guide on how to evaluate the polynomial expression x² + 5x + 1 when x = -2. We will delve into the process of substituting the given value, applying the order of operations, and simplifying the expression to arrive at the final answer. Whether you're a student learning algebra or simply seeking to refresh your math skills, this guide will provide clear and concise instructions. We'll break down each step, ensuring a solid understanding of the concepts involved. By following this detailed explanation, you'll gain confidence in your ability to evaluate similar expressions accurately and efficiently. Let's embark on this mathematical journey together and unlock the secrets of polynomial evaluation. This foundational skill is crucial for success in higher-level mathematics, including calculus and beyond. Mastering the art of expression evaluation empowers you to solve complex problems and understand intricate mathematical relationships. So, let's get started and transform this seemingly challenging task into a manageable and rewarding experience. Remember, practice is key to perfection, and this guide will serve as your trusted companion in your mathematical endeavors. Stay focused, follow the steps diligently, and you'll undoubtedly achieve proficiency in evaluating expressions. Now, let's dive into the specifics of our example and unravel the mystery of x² + 5x + 1 when x = -2.

Understanding the Basics

Before we jump into the evaluation, let's clarify some essential concepts. An expression in mathematics is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. In our case, x² + 5x + 1 is a polynomial expression. A variable, like x, represents an unknown value. When we evaluate an expression, we substitute a specific value for the variable and perform the indicated operations to find the numerical result. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for accurate evaluation. It dictates the sequence in which operations must be performed. This ensures that we arrive at the correct answer, regardless of who is solving the problem. Ignoring the order of operations can lead to incorrect results and a misunderstanding of the expression's true value. Therefore, it's imperative to adhere to PEMDAS meticulously. Let's reiterate the importance of each step in PEMDAS: First, we address operations within Parentheses. Next, we tackle Exponents. Then, we perform Multiplication and Division from left to right. Finally, we execute Addition and Subtraction from left to right. By consistently applying PEMDAS, we can confidently navigate the complexities of mathematical expressions and arrive at precise solutions. Understanding these fundamental principles lays the groundwork for successfully evaluating x² + 5x + 1 when x = -2. With a clear grasp of expressions, variables, evaluation, and the order of operations, we are well-equipped to tackle the problem at hand. Let's move forward and apply these concepts to our specific example, step by step.

Step-by-Step Evaluation of x² + 5x + 1 when x = -2

Now, let's proceed with the evaluation of the expression x² + 5x + 1 when x = -2. This will involve a series of carefully executed steps, each building upon the previous one. Our primary goal is to replace the variable x with the value -2 and then simplify the expression using the order of operations. This systematic approach will ensure accuracy and clarity in our solution. Remember, precision is paramount in mathematics, and each step must be performed with meticulous attention to detail. Let's begin by substituting -2 for x in the expression. This substitution is the cornerstone of the evaluation process, setting the stage for the subsequent calculations. Once we've made the substitution, we'll have a numerical expression that we can simplify using PEMDAS. The beauty of mathematics lies in its logical progression, where each step is a natural consequence of the preceding one. By adhering to this logical flow, we can confidently navigate complex problems and arrive at correct solutions. So, let's embark on this journey of evaluation, starting with the crucial substitution step.

1. Substitute x with -2:

The first step in evaluating the expression x² + 5x + 1 when x = -2 is to substitute every instance of x with the value -2. This is a crucial step because it transforms the algebraic expression into a numerical expression that we can then simplify. Be especially mindful of the negative sign when substituting. It's a common mistake to overlook the negative sign, which can lead to an incorrect answer. The substitution should be done carefully and precisely to ensure accuracy. This process involves replacing the variable x with the numerical value -2 in the original expression. It's like replacing a placeholder with a specific value. Once the substitution is complete, we'll have an expression consisting only of numbers and mathematical operations. This numerical expression is what we'll work with in the subsequent steps. The result of this substitution is: (-2)² + 5(-2) + 1. Notice how we've enclosed -2 in parentheses to maintain clarity and prevent confusion, especially when dealing with exponents and multiplication. This notation is crucial for avoiding errors and ensuring that the operations are performed in the correct order. Let's move forward with the next step, where we'll apply the order of operations to simplify this numerical expression.

2. Apply the Order of Operations (PEMDAS):

Now that we have substituted x with -2, our expression is (-2)² + 5(-2) + 1. We need to simplify this expression using the order of operations (PEMDAS). This ensures we perform the operations in the correct sequence to arrive at the accurate result. PEMDAS, as a reminder, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In our expression, we first encounter an exponent: (-2)². This means -2 multiplied by itself, which is (-2) * (-2) = 4. Remember that a negative number multiplied by a negative number results in a positive number. Next, we have multiplication: 5(-2). This means 5 multiplied by -2, which is -10. Now our expression looks like this: 4 + (-10) + 1. We are left with addition. We perform addition from left to right. First, we add 4 and -10, which gives us -6. Then, we add -6 and 1, which results in -5. Therefore, after applying the order of operations, we have simplified the expression to a single numerical value. This step-by-step simplification is the essence of evaluating mathematical expressions. By adhering to PEMDAS, we can navigate the complexities of operations and arrive at the correct answer. Let's move on to the final step, where we'll state the result of our evaluation.

3. Simplify the Expression:

Following the order of operations, we've simplified our expression step-by-step. Let's recap the process: We started with (-2)² + 5(-2) + 1. First, we evaluated the exponent: (-2)² = 4. Then, we performed the multiplication: 5(-2) = -10. This gave us the expression 4 + (-10) + 1. Next, we performed the addition from left to right: 4 + (-10) = -6. Finally, we added -6 and 1: -6 + 1 = -5. Therefore, the simplified value of the expression x² + 5x + 1 when x = -2 is -5. This is our final answer. The process of simplification is crucial in mathematics. It allows us to transform a complex expression into its most basic form, making it easier to understand and use. Each step in the simplification process brings us closer to the final solution. By carefully applying the order of operations and performing each calculation accurately, we can confidently arrive at the correct answer. In this case, we've successfully simplified the expression to -5, demonstrating our understanding of polynomial evaluation. Let's now formally state our conclusion, summarizing our findings and highlighting the key steps in our solution.

Conclusion

In conclusion, by substituting x = -2 into the expression x² + 5x + 1 and following the order of operations (PEMDAS), we have successfully evaluated the expression to be -5. This process involved three key steps: first, we substituted x with -2; second, we applied the order of operations to simplify the expression; and third, we arrived at the final answer of -5. This example demonstrates the importance of understanding and applying the fundamental principles of algebra, such as variable substitution and the order of operations. These skills are essential for success in more advanced mathematical topics. The ability to evaluate expressions accurately and efficiently is a cornerstone of mathematical proficiency. It allows us to solve equations, analyze functions, and model real-world phenomena. Therefore, mastering this skill is crucial for anyone pursuing a career in STEM fields or simply seeking to enhance their mathematical abilities. The process we've outlined in this article provides a clear and concise framework for evaluating polynomial expressions. By consistently following these steps, you can confidently tackle similar problems and achieve accurate results. Remember, practice is key to mastering any mathematical skill. So, we encourage you to try evaluating other expressions with different values of x to solidify your understanding. With dedication and consistent effort, you can develop your mathematical prowess and excel in your academic pursuits. This journey of mathematical exploration is filled with challenges and rewards. By embracing the challenges and celebrating the rewards, you can unlock your full potential and achieve your mathematical goals.

Final Answer: When x = -2, the expression x² + 5x + 1 evaluates to -5.

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Evaluate the expression x² + 5x + 1 when x equals -2.