Evaluate 2a^2 - 4b - 4a(b - A) Expression When A Is 4 And B Is 7
In this article, we will delve into the process of evaluating a given algebraic expression. Specifically, we will be focusing on the expression 2a^2 - 4b - 4a(b - a), where the values of the variables a and b are provided as a = 4 and b = 7. This type of problem is a fundamental concept in algebra, often encountered in introductory courses and standardized tests. Understanding how to correctly substitute values and simplify expressions is crucial for success in more advanced mathematical topics.
Understanding the Basics of Algebraic Expressions
Before we dive into the specific problem, let's quickly recap some fundamental concepts related to algebraic expressions. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. The order of operations (PEMDAS/BODMAS) is paramount when simplifying expressions: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
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), dictates the sequence in which operations must be performed. Failing to adhere to this order can lead to incorrect results. For instance, in the expression 2 + 3 * 4, we must perform the multiplication (3 * 4 = 12) before the addition (2 + 12 = 14), resulting in the correct answer of 14. If we were to add first (2 + 3 = 5) and then multiply (5 * 4 = 20), we would arrive at an incorrect answer. This underscores the vital role that the order of operations plays in achieving accurate simplification.
Substitution in Algebraic Expressions
Substitution is the process of replacing variables in an expression with their given values. This is a crucial step in evaluating algebraic expressions. Once the variables have been substituted, the expression becomes a numerical expression, which can then be simplified using the order of operations. Accurate substitution is key to obtaining the correct evaluation.
Let's Break Down the Given Expression
The expression we need to evaluate is 2a^2 - 4b - 4a(b - a). This expression contains several terms and operations. It includes terms with exponents (a^2), multiplication (2a^2, 4b, 4a(b - a)), subtraction, and parentheses. To evaluate this expression correctly, we will need to carefully substitute the given values for a and b and then simplify the resulting numerical expression using the order of operations.
Step-by-Step Evaluation of 2a^2 - 4b - 4a(b - a)
Now, let's proceed with the evaluation of the expression 2a^2 - 4b - 4a(b - a) given a = 4 and b = 7. We will break down the process into several steps to ensure clarity and accuracy.
Step 1: Substitute the Values of a and b
The first step is to substitute the given values of a and b into the expression. We replace a with 4 and b with 7:
2(4)^2 - 4(7) - 4(4)(7 - 4)
After substitution, the expression is now a numerical expression containing only numbers and operations. We can proceed to simplify it using the order of operations.
Step 2: Simplify the Expression Inside the Parentheses
According to the order of operations (PEMDAS/BODMAS), we must first address the expression within the parentheses. In this case, we have (7 - 4):
7 - 4 = 3
Now, we substitute this result back into the expression:
2(4)^2 - 4(7) - 4(4)(3)
Step 3: Evaluate the Exponent
The next operation in the order of operations is to evaluate the exponent. We have 4^2, which means 4 multiplied by itself:
4^2 = 4 * 4 = 16
Substituting this result back into the expression, we get:
2(16) - 4(7) - 4(4)(3)
Step 4: Perform the Multiplications
Now, we need to perform the multiplication operations from left to right. We have three multiplication operations:
- 2(16) = 32
- 4(7) = 28
- 4(4)(3) = 16(3) = 48
Substituting these results back into the expression, we have:
32 - 28 - 48
Step 5: Perform the Subtractions
Finally, we perform the subtraction operations from left to right:
- 32 - 28 = 4
- 4 - 48 = -44
Therefore, the final result of the evaluation is -44.
Conclusion: The Power of Order of Operations
In conclusion, the value of the expression 2a^2 - 4b - 4a(b - a) when a = 4 and b = 7 is -44. This result was obtained by meticulously following the order of operations and performing each step with precision. This exercise highlights the importance of adhering to the order of operations in simplifying algebraic expressions. Without a consistent and methodical approach, it is easy to make errors and arrive at an incorrect answer.
Importance of Practice in Algebra
Evaluating algebraic expressions is a fundamental skill in mathematics. By understanding and applying the order of operations, and practicing consistently, students can build a strong foundation for future success in mathematics. Practice is key to mastering these concepts. By working through various problems, one can develop a strong intuition for algebraic manipulations and problem-solving strategies.
Real-World Applications of Algebra
Algebra is not just an abstract concept confined to textbooks and classrooms. It has a multitude of real-world applications. From calculating the trajectory of a rocket to designing bridges and buildings, algebra plays a crucial role in various fields of science, engineering, and technology. Even in everyday life, we use algebraic concepts implicitly when budgeting, cooking, or planning a trip. This reinforces the importance of developing a strong understanding of algebraic principles.
Therefore, mastering the evaluation of algebraic expressions, like the one we tackled in this article, is not only an academic pursuit but also a valuable skill that can be applied in numerous practical scenarios. Continuous learning and practice are essential to develop proficiency in this area and unlock its full potential.
