Evaluate The Expression (x^2 + 4x + 8) / (x^2 - 8x - 8) For X = 4
#h1 Understanding the Problem
In this article, we will delve into the process of finding the value of a given algebraic expression when a specific value is assigned to the variable. This is a fundamental concept in algebra, and mastering it is crucial for solving more complex mathematical problems. Our specific task is to evaluate the expression (x^2 + 4x + 8) / (x^2 - 8x - 8) when x = 4. This involves substituting the value of x into the expression and simplifying the result. The expression involves polynomial terms in both the numerator and the denominator, making it a rational expression. The steps to solving this problem include substitution, arithmetic operations (addition, subtraction, multiplication, and division), and simplification. Paying close attention to the order of operations (PEMDAS/BODMAS) is essential to prevent errors. By following a systematic approach, we can accurately determine the value of the expression when x equals 4. This foundational skill extends to various fields, including engineering, physics, and computer science, making it a cornerstone of mathematical education. The ability to manipulate and evaluate expressions forms the backbone of algebraic reasoning, paving the way for problem-solving across a multitude of disciplines. So, let's embark on this step-by-step journey to unveil the solution.
Step-by-Step Solution
1. Substitution
The first step in solving this problem is to substitute the given value of x, which is 4, into the expression. The original expression is:
(x^2 + 4x + 8) / (x^2 - 8x - 8)
Replacing x with 4, we get:
(4^2 + 4(4) + 8) / (4^2 - 8(4) - 8)
This step is crucial as it sets the stage for the subsequent arithmetic operations. Careful substitution is essential to avoid errors that can propagate through the rest of the solution. The expression now contains only numerical values, allowing us to proceed with simplification using the order of operations.
2. Simplify the Numerator
Next, we simplify the numerator of the expression. The numerator is:
4^2 + 4(4) + 8
Following the order of operations (PEMDAS/BODMAS), we first evaluate the exponent:
4^2 = 16
Then, we perform the multiplication:
4(4) = 16
Now, we add the terms together:
16 + 16 + 8 = 40
So, the simplified numerator is 40. This step involves multiple arithmetic operations, and maintaining accuracy at each stage is vital. The numerator now has a single numerical value, making it easier to work with in the overall expression.
3. Simplify the Denominator
Now, we simplify the denominator of the expression. The denominator is:
4^2 - 8(4) - 8
Again, following the order of operations, we first evaluate the exponent:
4^2 = 16
Then, we perform the multiplication:
8(4) = 32
Now, we subtract the terms:
16 - 32 - 8
16 - 32 = -16
-16 - 8 = -24
So, the simplified denominator is -24. Similar to the numerator, careful execution of each arithmetic operation is essential to arrive at the correct value. The denominator now has a single numerical value, setting the stage for the final division.
4. Divide the Numerator by the Denominator
Now that we have simplified both the numerator and the denominator, we can divide the numerator by the denominator:
40 / -24
This fraction can be simplified by finding the greatest common divisor (GCD) of 40 and 24. The GCD of 40 and 24 is 8. Dividing both the numerator and the denominator by 8, we get:
(40 / 8) / (-24 / 8) = 5 / -3
So, the simplified fraction is -5/3. This division step is the final arithmetic operation, and ensuring accuracy is paramount. The simplified fraction represents the value of the original expression when x equals 4.
Final Answer
Therefore, the value of the expression (x^2 + 4x + 8) / (x^2 - 8x - 8) when x = 4 is -5/3. This result represents the culmination of the substitution, simplification, and division steps. The final answer is a simplified fraction, indicating the precise value of the expression under the given condition.
#h2 Why is this Important?
Understanding how to evaluate algebraic expressions is a foundational skill in mathematics with far-reaching applications. Evaluating expressions is not just a textbook exercise; it's a crucial tool for problem-solving in various real-world scenarios. The ability to substitute values into expressions and simplify them allows us to model and analyze relationships between different quantities.
Applications in Science and Engineering
In science and engineering, mathematical models are used to represent physical phenomena. These models often involve algebraic expressions that relate variables such as force, velocity, acceleration, and energy. For example, in physics, the equation for kinetic energy is KE = (1/2)mv^2, where KE is kinetic energy, m is mass, and v is velocity. To calculate the kinetic energy of an object, we need to substitute the values of mass and velocity into the expression and evaluate it. Similarly, in electrical engineering, Ohm's Law (V = IR) relates voltage (V), current (I), and resistance (R). Evaluating these expressions allows engineers to design circuits, predict their behavior, and troubleshoot issues. Understanding these concepts is crucial for students pursuing careers in STEM fields.
Applications in Economics and Finance
Economics and finance also rely heavily on algebraic expressions. Economists use models to analyze economic trends, predict market behavior, and make policy recommendations. For instance, the formula for compound interest, A = P(1 + r/n)^(nt), where A is the future value, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years, is a prime example of an algebraic expression used in finance. Evaluating this expression helps investors determine the potential return on their investments. Financial analysts use various algebraic expressions to assess risk, value assets, and make investment decisions. The ability to manipulate and evaluate these expressions is essential for anyone working in the financial sector.
Everyday Problem Solving
Beyond specialized fields, evaluating expressions is a valuable skill for everyday problem-solving. Whether you are calculating the total cost of items on sale, determining the tip amount at a restaurant, or figuring out the monthly payments on a loan, the ability to substitute values into formulas and simplify them is essential. For example, when calculating the total cost of an item with a discount, you might use the expression: Total Cost = Original Price - (Discount Rate * Original Price). Substituting the values for the original price and discount rate allows you to quickly determine the final cost. This skill empowers individuals to make informed decisions in their personal and professional lives.
Building a Foundation for Advanced Math
Mastering the evaluation of expressions also lays a strong foundation for more advanced mathematical concepts. As you progress in mathematics, you will encounter more complex algebraic expressions, equations, and functions. The ability to manipulate and evaluate expressions is a prerequisite for understanding these advanced topics. For example, solving equations involves simplifying expressions and isolating variables. Calculus, which deals with rates of change and accumulation, relies heavily on the ability to work with functions, which are essentially algebraic expressions. A solid understanding of the basics ensures a smoother transition to higher-level mathematics.
#h3 Common Mistakes to Avoid
When evaluating expressions, it's easy to make mistakes if you're not careful. Avoiding common errors can significantly improve your accuracy and efficiency in solving mathematical problems. Here are some common pitfalls to watch out for:
Incorrect Order of Operations
The most common mistake is failing to follow the correct order of operations (PEMDAS/BODMAS). This acronym stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Neglecting this order can lead to incorrect results. For example, in the expression 2 + 3 * 4, you must perform the multiplication before the addition. If you add 2 and 3 first, you'll get 5 * 4 = 20, which is incorrect. The correct answer is 2 + 12 = 14. Consistently applying PEMDAS/BODMAS is crucial for accurate calculations. Students should practice with various expressions to internalize this order and avoid common mistakes.
Sign Errors
Dealing with negative numbers can be tricky, and sign errors are a frequent source of mistakes. A negative sign in front of a term or a parenthesis can change the entire expression's value. For example, consider the expression 5 - (3 - 2). If you incorrectly distribute the negative sign, you might calculate 5 - 3 - 2 = 0, which is wrong. The correct way is 5 - (3 - 2) = 5 - 1 = 4. Always pay close attention to signs, and remember that subtracting a negative number is equivalent to adding its positive counterpart. Practice with expressions involving negative numbers helps to reinforce these rules and minimize errors.
Incorrect Substitution
Another common mistake is incorrect substitution. When substituting a value for a variable, ensure you replace every instance of the variable with the correct value. Also, be careful with parentheses, especially when substituting negative numbers. For example, if the expression is x^2 and x = -3, the correct substitution is (-3)^2, which equals 9. If you write -3^2, you might incorrectly calculate -9, as the exponent only applies to the 3, not the negative sign. Double-check your substitutions to prevent errors that can propagate through the rest of the problem.
Arithmetic Errors
Simple arithmetic errors, such as miscalculating multiplication or addition, can also lead to incorrect answers. These errors are often due to carelessness or rushing through the problem. For example, miscalculating 7 * 8 as 54 instead of 56 is a common arithmetic error. To minimize these errors, it's helpful to show your work step-by-step and double-check each calculation. Using a calculator for complex calculations can also reduce the chances of arithmetic mistakes. Accuracy in basic arithmetic operations is essential for solving mathematical problems correctly.
Forgetting to Simplify
Finally, forgetting to simplify the expression completely is another common mistake. After substituting and performing operations, make sure to reduce the expression to its simplest form. This might involve combining like terms, reducing fractions, or canceling out common factors. For example, if you end up with the fraction 6/8, you should simplify it to 3/4. Leaving the answer in an unsimplified form might not be considered fully correct. Always take the extra step to simplify your answer to ensure you've completed the problem accurately.
#h4 Practice Problems
To solidify your understanding of evaluating expressions, it's essential to practice with a variety of problems. The more you practice, the more comfortable and confident you'll become with the process. Here are a few practice problems to get you started:
- Evaluate 3x^2 - 2x + 5 when x = -2.
- Evaluate (2a + 3b) / (a - b) when a = 4 and b = 1.
- Evaluate √((x^2 + y^2)) when x = 3 and y = 4.
- Evaluate (m^3 - 4m + 2) / (m + 1) when m = 2.
- Evaluate 5(p - q)^2 + 3p when p = 5 and q = 2.
#h5 Conclusion
In conclusion, finding the value of an expression by substituting a given value for the variable is a fundamental skill in mathematics. In this article, we have demonstrated the step-by-step process of evaluating the expression (x^2 + 4x + 8) / (x^2 - 8x - 8) when x = 4. The solution involves substituting the value, simplifying the numerator and denominator, and performing the final division. Mastering this process is crucial for success in algebra and beyond. Remember to always follow the order of operations, pay attention to signs, and simplify your answer completely. With practice and attention to detail, you can confidently tackle any expression evaluation problem. This skill not only enhances your mathematical abilities but also provides a valuable tool for problem-solving in various real-world applications.
