Evaluate 3(x-1)^2+2x-7 For X=3 Step-by-Step Guide
In mathematics, evaluating expressions is a fundamental skill. This article will guide you through the process of evaluating the expression 3(x-1)^2 + 2x - 7 when x = 3. We will break down each step, providing clear explanations and insights to help you master this essential concept. This comprehensive guide ensures that you not only understand the mechanics of solving the problem but also grasp the underlying principles of algebraic evaluation.
Understanding the Expression
Before we dive into the solution, let’s take a moment to understand the expression we are working with: 3(x-1)^2 + 2x - 7. This algebraic expression involves several mathematical operations, including multiplication, subtraction, exponentiation, and addition. The variable x represents a placeholder for a numerical value, and our goal is to find the value of the entire expression when we substitute x with the number 3. Recognizing the order of operations is crucial here. We need to adhere to the PEMDAS/BODMAS rule, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This rule dictates the sequence in which we perform the operations to arrive at the correct answer. Misinterpreting this order can lead to significant errors, so understanding and applying it correctly is paramount.
Step 1: Substituting the Value of x
The first step in evaluating the expression is to substitute the value of x with 3. This means we replace every instance of x in the expression with the number 3. So, the expression 3(x-1)^2 + 2x - 7 becomes 3(3-1)^2 + 2(3) - 7. Substitution is a fundamental concept in algebra, and it's the cornerstone of evaluating expressions and solving equations. It allows us to replace variables with specific numerical values, transforming abstract expressions into concrete calculations. Accuracy in this step is crucial, as any mistake here will propagate through the rest of the solution. Double-checking the substitution ensures that you are working with the correct numerical expression.
Step 2: Simplifying Inside the Parentheses
Following the order of operations (PEMDAS/BODMAS), we first need to simplify the expression inside the parentheses. In our case, we have (3-1) inside the parentheses, which simplifies to 2. So, the expression now becomes 3(2)^2 + 2(3) - 7. Simplifying within parentheses is a critical step because it streamlines the expression and prepares it for subsequent operations. By addressing the parentheses first, we reduce the complexity of the expression, making it easier to manage and evaluate. This step highlights the importance of adhering to the correct order of operations to avoid errors and maintain the integrity of the mathematical process.
Step 3: Evaluating the Exponent
Next, we need to evaluate the exponent. We have 2^2, which means 2 raised to the power of 2, or 2 multiplied by itself. This equals 4. The expression now looks like this: 3(4) + 2(3) - 7. Exponents indicate repeated multiplication, and their evaluation is a key component of simplifying expressions. Understanding how to correctly calculate exponents is vital for accurate mathematical computations. In this case, squaring the number 2 results in 4, which then becomes a crucial value in the subsequent steps of our evaluation.
Step 4: Performing Multiplication
Now, we perform the multiplication operations from left to right. We have two multiplication operations: 3(4) and 2(3). 3(4) equals 12, and 2(3) equals 6. Our expression is now: 12 + 6 - 7. Multiplication is a fundamental arithmetic operation, and performing it correctly in the context of an algebraic expression is crucial. By executing the multiplication operations in the correct order, we continue to simplify the expression, bringing us closer to the final solution. This step reinforces the importance of adhering to the order of operations to maintain accuracy and logical flow.
Step 5: Performing Addition and Subtraction
Finally, we perform the addition and subtraction operations from left to right. We have 12 + 6 - 7. First, we add 12 and 6, which gives us 18. Then, we subtract 7 from 18, which results in 11. Therefore, the final value of the expression is 11. Addition and subtraction are the final steps in our evaluation process, and performing them in the correct sequence ensures that we arrive at the accurate result. By adding and subtracting from left to right, we adhere to the order of operations and complete the simplification of the expression. The final value of 11 represents the solution to our initial problem, demonstrating the power and precision of algebraic evaluation.
Final Answer
Therefore, the value of the expression 3(x-1)^2 + 2x - 7 when x = 3 is 11. This step-by-step evaluation demonstrates the importance of following the order of operations and carefully executing each step. By breaking down the problem into smaller, manageable parts, we can confidently arrive at the correct solution. This process not only provides the answer but also reinforces the fundamental principles of algebraic manipulation and evaluation.
Conclusion
Evaluating expressions is a core skill in mathematics. By following the order of operations and breaking down the problem into smaller steps, you can confidently evaluate even complex expressions. This detailed guide has provided a comprehensive walkthrough of evaluating 3(x-1)^2 + 2x - 7 for x = 3, emphasizing the importance of each step and the underlying mathematical principles. Mastering these skills will empower you to tackle more advanced mathematical problems with ease and precision.
First Step in Evaluating the Expression
The first step in evaluating the expression 3(x-1)^2 + 2x - 7 is to substitute the value of x. This means replacing every instance of x in the expression with the given value, which in this case is 3. So, the first step transforms the expression 3(x-1)^2 + 2x - 7 into 3(3-1)^2 + 2(3) - 7. This initial substitution is crucial because it converts the algebraic expression, which contains a variable, into a numerical expression that can be simplified using arithmetic operations. Without this first step, it would be impossible to proceed with the evaluation. The accuracy of this substitution is paramount, as any error made here will propagate through the rest of the calculation, leading to an incorrect final answer. Therefore, it is essential to double-check the substitution to ensure that the numerical expression is set up correctly for the subsequent steps.
Why Substitution is the First Step
Substitution serves as the foundational step in evaluating algebraic expressions because it bridges the gap between the abstract symbolic representation and concrete numerical calculation. By replacing the variable x with its specific value, we are essentially defining the context in which the expression is to be evaluated. This process allows us to move from a general form to a specific instance, making it possible to apply the rules of arithmetic to simplify and find the numerical result. The act of substitution is not merely a mechanical replacement; it is a crucial cognitive step that sets the stage for the entire evaluation process. It requires understanding the role of variables in algebraic expressions and the ability to translate abstract symbols into tangible numbers. This foundational understanding is critical for building more advanced mathematical skills.
The Importance of Accurate Substitution
The significance of accurate substitution cannot be overstated. A single error in this initial step can lead to a completely incorrect final answer. For example, if we were to mistakenly substitute x with a different number or misplace a digit, the subsequent calculations would be based on a flawed foundation. This is why meticulous attention to detail is essential during the substitution process. It involves not only correctly identifying each instance of the variable x but also ensuring that the replacement value is accurately inserted into the expression. This level of precision is a hallmark of careful mathematical practice and is crucial for achieving reliable results. By prioritizing accuracy in the substitution step, we lay the groundwork for a successful evaluation and avoid potential pitfalls that could compromise the final solution.
Substitution in More Complex Expressions
While the expression 3(x-1)^2 + 2x - 7 is relatively straightforward, the principle of substitution applies equally to more complex algebraic expressions. Whether dealing with multiple variables, nested parentheses, or intricate combinations of operations, the first step remains the same: substitute the given values for the variables. This consistent approach provides a reliable method for tackling a wide range of mathematical problems. As expressions become more complex, the importance of careful substitution only increases. It requires a systematic approach to ensure that each variable is correctly replaced, and the resulting numerical expression accurately reflects the original algebraic form. This ability to handle substitution in complex scenarios is a key skill in advanced algebra and beyond.
Real-World Applications of Evaluating Expressions
Evaluating expressions is not just an abstract mathematical exercise; it has numerous real-world applications. From calculating financial returns to modeling physical phenomena, the ability to substitute values into formulas and expressions is essential in many fields. For example, in physics, one might use an expression to calculate the trajectory of a projectile, substituting values for initial velocity, angle, and gravitational acceleration. In finance, expressions are used to calculate interest rates, loan payments, and investment growth. The common thread in these applications is the need to evaluate an expression given specific input values. This highlights the practical importance of mastering the skill of substitution and expression evaluation. It is a foundational tool that enables us to make informed decisions, solve real-world problems, and understand the quantitative aspects of our world.
In conclusion, the first step in evaluating the expression 3(x-1)^2 + 2x - 7 for x = 3 is to substitute the value of x with 3, transforming the expression into 3(3-1)^2 + 2(3) - 7. This step is fundamental, setting the stage for the rest of the evaluation process, and its accuracy is crucial for obtaining the correct final answer. By understanding the importance of substitution and practicing it diligently, you can build a solid foundation for more advanced mathematical concepts and applications.
