Evaluating 3x^2 - 6x + 7 For X = 7 A Step-by-Step Guide

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In mathematics, evaluating expressions is a fundamental skill. It involves substituting a given value for a variable and performing the necessary arithmetic operations to find the result. This article focuses on evaluating the expression 3x² - 6x + 7 when x = 7. This is a quintessential example of how algebraic expressions are handled in basic algebra and is a stepping stone to more complex mathematical problems. By understanding the process involved in this evaluation, readers can build a solid foundation in algebra and enhance their problem-solving abilities.

Understanding the Expression

Before we dive into the evaluation, let's break down the expression 3x² - 6x + 7. This is a quadratic expression, meaning it includes a term with the variable raised to the power of 2 (x²). The expression also includes a linear term (-6x) and a constant term (+7). Each term plays a crucial role in determining the overall value of the expression for a given value of x. Understanding the structure of the expression is the first step towards accurate evaluation. It’s essential to recognize the order of operations (PEMDAS/BODMAS) to ensure the expression is evaluated correctly. PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Applying this order is critical in obtaining the correct result.

Breaking Down the Terms

The expression 3x² - 6x + 7 consists of three distinct terms, each contributing uniquely to the final value. Let's analyze each term individually:

  1. 3x²: This is the quadratic term, where x is squared and then multiplied by 3. The coefficient 3 scales the value of x², making it a significant component of the expression, especially as x increases. The exponent indicates that x is multiplied by itself, and this result is then multiplied by the coefficient.
  2. -6x: This is the linear term, where x is multiplied by -6. This term introduces a negative component that changes linearly with x. The coefficient -6 indicates the rate at which the term changes with respect to x. Understanding linear terms is crucial as they represent direct relationships in many real-world scenarios.
  3. +7: This is the constant term, which remains unchanged regardless of the value of x. It provides a fixed value that is added to the result of the other terms. Constant terms are essential as they set the base value of the expression.

Understanding each term's role helps in predicting how the expression will behave for different values of x. For example, the quadratic term will dominate for large values of x, while the constant term will be more significant for values close to zero.

The Importance of Order of Operations

The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is crucial for evaluating mathematical expressions correctly. This set of rules ensures that everyone arrives at the same answer, regardless of who is performing the calculation. In the context of the expression 3x² - 6x + 7, following the order of operations is paramount. Here’s how it applies:

  1. Exponents: First, we must address any exponents. In this case, x² means x raised to the power of 2, so we calculate the square of x before anything else.
  2. Multiplication and Division: Next, we perform any multiplication or division from left to right. Here, we have 3 multiplied by x² and -6 multiplied by x. These multiplications must be done before any addition or subtraction.
  3. Addition and Subtraction: Finally, we carry out any addition or subtraction from left to right. In this expression, we subtract the result of -6x from the result of 3x², and then add 7.

By strictly adhering to the order of operations, we ensure that the expression is evaluated accurately. For instance, if we were to multiply before handling the exponent, we would arrive at an incorrect result. The order of operations is not just a mathematical convention; it is a fundamental principle that underpins all algebraic manipulations and calculations.

Substituting x = 7

Now that we understand the expression, we can substitute x = 7 into it. This means replacing every instance of x in the expression with the number 7. The expression then becomes 3(7)² - 6(7) + 7. This substitution is the critical first step in evaluating the expression for a specific value. Accurate substitution is essential to prevent errors and ensure the correct result. Each x must be replaced with the given value, maintaining the integrity of the expression's structure.

Performing the Substitution

Substituting x = 7 into the expression 3x² - 6x + 7 involves replacing each occurrence of x with the value 7. This yields the new expression 3(7)² - 6(7) + 7. The parentheses are used to clearly indicate multiplication, which is crucial for following the order of operations correctly. This step is straightforward but requires careful attention to detail to avoid mistakes. Ensuring that each x is correctly replaced with 7 sets the stage for the next steps in the evaluation process.

  1. Replacing x: Each x in the original expression is replaced by 7. This is a direct substitution, transforming the algebraic expression into an arithmetic expression.
  2. Maintaining the Structure: The structure of the expression remains the same, with only the variable x being replaced by its numerical value. This ensures that the subsequent calculations follow the correct mathematical relationships.
  3. Using Parentheses: Parentheses are used to indicate multiplication, especially when a number is multiplied by an expression within parentheses. This helps to avoid confusion and ensures that the correct operations are performed in the right order.

By meticulously performing the substitution, we prepare the expression for the next phase, which involves applying the order of operations to simplify and find the final value.

Applying the Order of Operations

With x substituted, we now evaluate 3(7)² - 6(7) + 7 using the order of operations (PEMDAS/BODMAS). First, we handle the exponent, then multiplication, and finally addition and subtraction. This systematic approach ensures we arrive at the correct answer. The order of operations is not just a guideline but a strict set of rules that must be followed to ensure accuracy. Neglecting this order can lead to incorrect results, highlighting the importance of a methodical approach to mathematical evaluations.

Step-by-Step Evaluation

Following the order of operations (PEMDAS/BODMAS), we evaluate the expression 3(7)² - 6(7) + 7 step by step:

  1. Exponent: The first step is to evaluate the exponent. We have 7², which means 7 multiplied by itself: 7² = 7 * 7 = 49. This simplifies the expression to 3(49) - 6(7) + 7.
  2. Multiplication: Next, we perform the multiplications from left to right. We have two multiplication operations: 3(49) and -6(7). Let's calculate these:
    • 3(49) = 147
    • -6(7) = -42 The expression now becomes 147 - 42 + 7.
  3. Addition and Subtraction: Finally, we perform the addition and subtraction from left to right:
    • 147 - 42 = 105
    • 105 + 7 = 112

Therefore, the final value of the expression is 112. Each step is crucial and builds upon the previous one, ensuring a clear and accurate path to the solution. This methodical approach not only provides the correct answer but also reinforces the importance of following established mathematical principles.

The Final Result

After following the order of operations, we find that 3(7)² - 6(7) + 7 = 112. Therefore, the value of the expression 3x² - 6x + 7 when x = 7 is 112. This result is the culmination of careful substitution and methodical evaluation. It demonstrates the power of algebraic expressions to represent numerical values under specific conditions. This final answer is not just a number; it is the result of applying mathematical principles to solve a problem, highlighting the practical application of algebraic concepts.

Verifying the Result

To ensure the accuracy of our result, it is always a good practice to verify the solution. There are several ways to verify the result of evaluating an expression:

  1. Recalculation: One of the simplest methods is to recalculate the expression, retracing each step to ensure no errors were made. This involves re-performing the substitution and following the order of operations again.
  2. Using a Calculator: A calculator can be used to directly evaluate the expression. By inputting the expression 3(7)² - 6(7) + 7 into a calculator, we can quickly check if the result matches our calculated value.
  3. Alternative Methods: Sometimes, there are alternative methods to evaluate the expression. For example, we could factor the expression or use a different algebraic manipulation to arrive at the same result.
  4. Software Verification: Mathematical software or online tools can be used to verify the result. These tools can handle complex calculations and provide an accurate evaluation of the expression.

By employing one or more of these verification methods, we can confidently confirm that the final result of 112 is correct. This step is crucial in ensuring the reliability of our solution and reinforcing the importance of accuracy in mathematical calculations.

Conclusion

Evaluating the expression 3x² - 6x + 7 for x = 7 demonstrates a fundamental algebraic skill. By substituting the value of x and following the order of operations, we arrived at the result of 112. This process underscores the importance of precision and attention to detail in mathematics. The ability to evaluate expressions is a crucial building block for more advanced mathematical concepts. It is a skill that is used extensively in various fields, from engineering to economics. Mastering this skill enhances problem-solving capabilities and provides a solid foundation for further mathematical studies. The exercise of evaluating 3x² - 6x + 7 for x = 7 is more than just a mathematical calculation; it is a practice in logical thinking and methodical problem-solving. The steps involved – understanding the expression, substituting the value, applying the order of operations, and verifying the result – are all essential components of mathematical proficiency. By consistently practicing these steps, individuals can develop a strong aptitude for algebra and excel in more complex mathematical challenges. This example serves as a reminder that mathematics is not just about memorizing formulas but about understanding the underlying principles and applying them effectively.