Evaluating Algebraic Expressions 2x^2 + 3(x + X) + 4 For X = 2

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Evaluating algebraic expressions is a fundamental skill in mathematics. It involves substituting given values for variables and performing the indicated operations to find the numerical result. In this article, we will delve into the process of evaluating the expression 2x^2 + 3(x + x) + 4 when x = 2. This step-by-step guide will not only provide the solution but also enhance your understanding of the order of operations and algebraic simplification. Whether you're a student learning algebra or someone looking to refresh your mathematical skills, this comprehensive explanation will be beneficial. So, let's embark on this mathematical journey together and unravel the value of this expression.

Step-by-Step Evaluation

1. Substitution

The first critical step in evaluating an algebraic expression is substitution. This involves replacing the variable, in this case, 'x', with its given value, which is 2. By substituting the value, we transform the algebraic expression into a numerical expression that we can then simplify. Careful substitution is crucial because any error in this initial step can lead to an incorrect final answer. It sets the foundation for the rest of the evaluation process, ensuring that we are working with the correct numerical values. Let's take a closer look at how this substitution is applied to our expression:

Original expression: 2x^2 + 3(x + x) + 4

Substituting x = 2:

2(2)^2 + 3(2 + 2) + 4

Notice how every instance of 'x' in the original expression has been replaced with the number 2. The parentheses are crucial here, as they maintain the correct order of operations. The next steps will involve simplifying this numerical expression, following the rules of arithmetic.

2. Simplifying Inside Parentheses

After the substitution, the next crucial step is to simplify the expressions within parentheses. Parentheses act as grouping symbols in mathematics, indicating that the operations inside them should be performed before any other operations outside. This rule is a fundamental part of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By simplifying inside the parentheses first, we maintain the correct order and avoid potential errors in our calculation. In our expression, we have a set of parentheses containing the addition operation, which must be addressed before proceeding further. Let's examine the simplification process within the parentheses in detail:

Our expression after substitution is: 2(2)^2 + 3(2 + 2) + 4

Now, we focus on the term inside the parentheses: (2 + 2)

Performing the addition:

2 + 2 = 4

So, the expression becomes:

2(2)^2 + 3(4) + 4

By simplifying the expression within the parentheses, we have reduced the complexity of the overall expression and paved the way for the next steps in the evaluation process. The next operation we will tackle is exponents, as dictated by the order of operations.

3. Evaluating Exponents

Once we've simplified the expressions within parentheses, the next priority is evaluating exponents. Exponents represent repeated multiplication, and they play a crucial role in many mathematical expressions. According to the order of operations (PEMDAS), exponents should be dealt with before multiplication, division, addition, or subtraction. This ensures that we calculate the powers correctly before performing other operations. In our expression, we have one term with an exponent, which is (2)^2. This term signifies that we need to multiply 2 by itself. Let's delve into the evaluation of this exponent:

Our expression after simplifying the parentheses is: 2(2)^2 + 3(4) + 4

Now, we focus on the term with the exponent: (2)^2

Evaluating the exponent:

(2)^2 = 2 * 2 = 4

Substituting the result back into the expression, we get:

2(4) + 3(4) + 4

By evaluating the exponent, we have further simplified our expression. We've transformed the exponential term into a simple numerical value, making it easier to proceed with the remaining operations. The next steps will involve handling multiplication and then addition, as we continue to follow the order of operations to reach our final answer.

4. Performing Multiplication

With the exponents evaluated, the next step in simplifying our expression is performing the multiplication operations. Multiplication is a fundamental arithmetic operation that involves combining groups of equal sizes. In the order of operations (PEMDAS), multiplication and division are performed before addition and subtraction. This means we need to address all multiplication instances in our expression before moving on to addition. Our expression currently has two multiplication operations: 2(4) and 3(4). Let's break down how we handle each of these multiplications:

Our expression after evaluating exponents is: 2(4) + 3(4) + 4

Now, we focus on the multiplication operations:

First multiplication: 2(4)

Performing the multiplication:

2 * 4 = 8

Second multiplication: 3(4)

Performing the multiplication:

3 * 4 = 12

Substituting these results back into the expression, we get:

8 + 12 + 4

By performing the multiplication operations, we have significantly simplified the expression. We've reduced the number of operations required to reach the final answer, and we are now left with only addition operations. The next and final step will involve adding the remaining terms together to find the value of the expression.

5. Performing Addition

After handling all multiplication, the final step in evaluating our expression is to perform the addition. Addition is one of the basic arithmetic operations, and in the order of operations (PEMDAS), it is typically the last operation to be carried out after parentheses, exponents, multiplication, and division. In our simplified expression, we are left with a series of addition operations: 8 + 12 + 4. To find the final value of the expression, we need to add these numbers together. Let's go through the addition process step by step:

Our expression after performing multiplication is: 8 + 12 + 4

Now, we perform the addition operations:

First addition: 8 + 12

Performing the addition:

8 + 12 = 20

Now, we add the result to the remaining term:

20 + 4

Performing the addition:

20 + 4 = 24

So, the final result of the expression is:

24

By performing the addition, we have successfully evaluated the entire expression. We have combined all the remaining terms to arrive at a single numerical value, which represents the solution to our problem. This completes the evaluation process, and we can confidently state the final answer.

Final Answer

After carefully following each step of the evaluation process, we have arrived at the final answer. We began by substituting the value of x into the expression, then simplified within parentheses, evaluated exponents, performed multiplication, and finally, carried out the addition. Each step was crucial in ensuring the accuracy of our result. The systematic approach, guided by the order of operations, allowed us to break down the complex expression into manageable parts and arrive at the solution. Now, let's state the final answer to our evaluation problem:

The value of the expression 2x^2 + 3(x + x) + 4 when x = 2 is:

24

This final answer represents the culmination of all the steps we have taken. It is the numerical value that the expression takes on when the variable x is replaced with 2. This result not only answers the specific problem we set out to solve but also reinforces the importance of following the order of operations and the principles of algebraic evaluation.

Conclusion

In conclusion, evaluating algebraic expressions is a fundamental skill in mathematics that requires a systematic approach and a solid understanding of the order of operations. In this article, we successfully evaluated the expression 2x^2 + 3(x + x) + 4 when x = 2. By meticulously following each step – substitution, simplification within parentheses, evaluation of exponents, multiplication, and addition – we arrived at the final answer of 24. This process highlights the significance of PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) in ensuring accurate calculations.

Furthermore, this exercise demonstrates the importance of breaking down complex problems into smaller, more manageable steps. By addressing each operation in the correct sequence, we can simplify even the most intricate expressions. This step-by-step methodology is not only applicable to algebraic evaluations but also to various problem-solving scenarios in mathematics and beyond.

Understanding how to evaluate expressions is crucial for further studies in algebra and calculus. It forms the basis for solving equations, graphing functions, and tackling more advanced mathematical concepts. Therefore, mastering this skill is an essential investment in your mathematical journey. Whether you are a student, a professional, or simply someone with an interest in mathematics, the ability to confidently evaluate expressions will undoubtedly prove to be a valuable asset.

Summary of Steps

To recap, here's a concise summary of the steps we took to evaluate the expression:

  1. Substitution: Replace the variable x with its given value, 2.
  2. Simplifying Inside Parentheses: Perform the addition within the parentheses: (2 + 2) = 4.
  3. Evaluating Exponents: Calculate the value of the exponent: (2)^2 = 4.
  4. Performing Multiplication: Multiply the terms: 2(4) = 8 and 3(4) = 12.
  5. Performing Addition: Add the remaining terms: 8 + 12 + 4 = 24.

By following these steps, we systematically simplified the expression and arrived at the correct answer. This structured approach is key to success in evaluating algebraic expressions.

Practice Problems

To further enhance your understanding and skills in evaluating expressions, try solving these practice problems:

  1. Evaluate 3x^2 - 2(x + 1) + 5 when x = 3.
  2. Evaluate (4x - 1)^2 + 2x when x = 1.
  3. Evaluate 5(x^2 + x) - 3 when x = -2.

Working through these problems will provide you with valuable practice and solidify your grasp of the concepts discussed in this article. Remember to follow the order of operations and take each step carefully. Happy calculating!