Evaluate (x^2 + 4x + 8) / (x^2 - 8x - 8) When X = 4

In this article, we will walk through the process of evaluating a rational expression for a given value of the variable. Specifically, we will focus on the expression $\frac{x^2+4 x+8}{x^2-8 x-8}$ and determine its value when $x=4$. This exercise is crucial for understanding algebraic manipulation and substitution, which are fundamental concepts in mathematics.

Understanding the Expression

Before we dive into the substitution, let's take a closer look at the expression itself. We have a rational expression, which is simply a fraction where the numerator and denominator are both polynomials. In this case, the numerator is $x^2 + 4x + 8$ and the denominator is $x^2 - 8x - 8$. Rational expressions are ubiquitous in algebra and calculus, appearing in various contexts such as solving equations, graphing functions, and simplifying complex expressions. To evaluate this expression, we need to replace the variable x with the given value, which in our case is 4, and then simplify the resulting numerical expression. This process will reveal the expression's value at that particular point. The importance of correctly substituting and simplifying cannot be overstated, as errors in these steps can lead to incorrect results. Understanding the structure of the expression and the order of operations is critical for accurate evaluation.

Step-by-Step Evaluation

Now, let's proceed with evaluating the expression step by step. The first step involves substituting $x=4$ into the expression. This means replacing every instance of x in the expression with the number 4. So, we have:

$\frac{4^2 + 4(4) + 8}{4^2 - 8(4) - 8}$

Next, we need to simplify the numerator and the denominator separately. Remember to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In the numerator, we have $4^2$, which is 16. Then, we have $4(4)$, which is also 16. So the numerator becomes $16 + 16 + 8$. In the denominator, we also have $4^2$, which is 16. Then, we have $8(4)$, which is 32. So the denominator becomes $16 - 32 - 8$. After performing these calculations, we have:

Numerator: $16 + 16 + 8 = 40$

Denominator: $16 - 32 - 8 = -24$

So, our expression now looks like this:

$\frac{40}{-24}$

The final step is to simplify this fraction. Both 40 and -24 are divisible by 8. Dividing both the numerator and the denominator by 8, we get:

$\frac{40 \div 8}{-24 \div 8} = \frac{5}{-3}$

Therefore, the value of the expression when $x=4$ is $-\frac{5}{3}$.

Common Mistakes to Avoid

When evaluating expressions, it's easy to make mistakes if you're not careful. One common mistake is not following the order of operations correctly. For instance, some might add 4 and 8 in the numerator before performing the multiplication, leading to an incorrect result. Another common mistake is mishandling negative signs, particularly in the denominator. For example, forgetting the negative sign when multiplying or subtracting can significantly alter the outcome. Also, be cautious when simplifying fractions. Ensure you're dividing both the numerator and the denominator by the greatest common divisor to obtain the simplest form. Double-checking your work at each step is always a good practice to catch these errors. It’s crucial to break down the expression into manageable parts and solve it step by step to minimize the chances of mistakes. Furthermore, practice is key to mastering these skills; the more you evaluate expressions, the more comfortable and accurate you'll become.

Importance of Substitution and Simplification

The process of substitution and simplification is not just a mathematical exercise; it's a fundamental skill that's crucial in various fields, including engineering, physics, computer science, and economics. In engineering, for example, you might need to substitute values into a formula to calculate stress on a bridge or the flow rate in a pipe. In physics, you might use substitution to determine the trajectory of a projectile or the energy of a particle. In computer science, it's used extensively in algorithms and programming to assign values to variables and evaluate conditions. Even in economics, substituting values into equations is essential for making predictions and analyzing market trends. The ability to accurately substitute values and simplify expressions is, therefore, a cornerstone of problem-solving in many disciplines. This skill provides a structured approach to tackle complex problems, making it an indispensable tool for anyone pursuing a career in these fields. It allows for a systematic evaluation of relationships between variables, enabling informed decision-making and accurate predictions.

Alternative Approaches and Further Exploration

While the step-by-step approach we used is the most straightforward for evaluating the expression, there might be alternative methods or scenarios to consider. For instance, if the expression could be simplified algebraically before substituting $x=4$, it might lead to a more streamlined calculation. However, in this specific case, the expression doesn't easily lend itself to algebraic simplification. Another avenue for exploration is to analyze the behavior of the expression for different values of x. You could graph the rational function $y = \frac{x^2+4 x+8}{x^2-8 x-8}$ to visualize how the value of the expression changes as x varies. You might also investigate the values of x for which the expression is undefined (i.e., where the denominator is zero). This could provide a deeper understanding of the expression's properties and limitations. Furthermore, one could explore similar expressions with different coefficients and observe how those changes impact the final value. Such investigations not only reinforce the process of substitution and simplification but also foster a deeper appreciation for the intricacies of mathematical expressions.

Conclusion

In conclusion, evaluating the expression $\frac{x^2+4 x+8}{x^2-8 x-8}$ when $x=4$ involves substituting the value of x into the expression and simplifying the result. By following the order of operations and carefully performing each step, we found that the value of the expression is $-\frac{5}{3}$. This exercise highlights the importance of substitution and simplification, which are crucial skills in mathematics and various other fields. Remember to avoid common mistakes by carefully following the order of operations and double-checking your work. Practicing these skills will undoubtedly enhance your mathematical proficiency and problem-solving abilities. This process not only yields a specific numerical answer but also reinforces a fundamental understanding of how variables interact within mathematical expressions. The ability to evaluate expressions accurately is a vital tool in any quantitative discipline, empowering individuals to make informed decisions and solve complex problems with confidence.