Finding Values For Zero Rational Expressions How To Solve (x-9)/((x-4)(x+4)) = 0

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In mathematics, rational expressions play a crucial role, especially when dealing with algebraic equations and functions. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. Understanding when these expressions equal zero is fundamental for solving equations, finding intercepts, and analyzing functions. This article delves into the process of determining the value of x that makes a given rational expression equal to zero, providing a detailed explanation and step-by-step solution. We will focus on the rational expression $\frac{x-9}{(x-4)(x+4)}$ and explore the conditions under which it becomes zero. By the end of this discussion, you'll have a solid grasp of the underlying principles and techniques involved in solving such problems.

Understanding Rational Expressions

Before diving into the specific problem, let's clarify what rational expressions are and why they are significant. A rational expression is a fraction where both the numerator and the denominator are polynomials. For example, expressions like $\frac{x^2 + 1}{x - 2}$ and $\frac{3x}{x^2 + 4x + 3}$ are rational expressions. These expressions are ubiquitous in algebra and calculus, appearing in various contexts, such as solving equations, graphing functions, and analyzing limits.

The key feature of a rational expression is that it is undefined when the denominator equals zero. This is because division by zero is undefined in mathematics. Therefore, when working with rational expressions, it's crucial to identify the values of x that make the denominator zero. These values are called excluded values or singularities because they are not part of the domain of the expression.

To find when a rational expression equals zero, we focus on the numerator. A fraction is equal to zero if and only if its numerator is equal to zero and its denominator is not zero. This is because zero divided by any non-zero number is zero. In contrast, if the denominator is also zero, the expression is undefined, not zero. Understanding this distinction is critical for accurately solving problems involving rational expressions.

In summary, the process of finding the values of x that make a rational expression zero involves two main steps: setting the numerator equal to zero and solving for x, and then checking that these values do not make the denominator equal to zero. If a value makes both the numerator and the denominator zero, it is not a solution; it's an excluded value.

Problem Statement: When is the Expression Zero?

Now, let's address the specific problem at hand. We are given the rational expression $\frac{x-9}{(x-4)(x+4)}$ and asked to find the value of x for which this expression equals zero. To solve this, we need to follow the principles outlined earlier: set the numerator equal to zero and ensure that the denominator is not simultaneously zero.

The numerator of the given expression is x - 9. Setting this equal to zero gives us the equation:

x−9=0x - 9 = 0

Solving for x, we add 9 to both sides of the equation:

x=9x = 9

So, the numerator is zero when x = 9. Now, we need to check if this value makes the denominator zero as well. The denominator of the expression is (x - 4)(x + 4). Let's substitute x = 9 into the denominator:

(9−4)(9+4)=(5)(13)=65(9 - 4)(9 + 4) = (5)(13) = 65

Since the denominator is 65 when x = 9, which is not zero, we can conclude that x = 9 is indeed a valid solution.

This step is crucial because if x = 9 also made the denominator zero, we would have an indeterminate form (0/0), which means the expression is undefined at that point, not zero. By verifying that the denominator is non-zero, we confirm that x = 9 is the value that makes the given rational expression equal to zero.

Step-by-Step Solution

To further illustrate the process, let's break down the solution into a step-by-step format:

  1. Identify the Numerator and Denominator:

    • Numerator: x - 9
    • Denominator: (x - 4)(x + 4)

  2. Set the Numerator Equal to Zero:

    • x - 9 = 0

  3. Solve for x:

    • Add 9 to both sides:
      • x = 9

  4. Check the Denominator:

    • Substitute x = 9 into the denominator (x - 4)(x + 4):
      • (9 - 4)(9 + 4) = (5)(13) = 65

  5. Verify the Solution:

    • Since the denominator is not zero when x = 9, the solution is valid.

Therefore, the value of x that makes the rational expression $\frac{x-9}{(x-4)(x+4)}$ equal to zero is x = 9. This step-by-step approach ensures a clear and systematic solution, reducing the likelihood of errors.

By following these steps, you can confidently solve similar problems involving rational expressions. The key is to focus on the numerator and denominator separately and always verify that your solution does not make the denominator zero.

Checking for Excluded Values

While finding the value of x that makes the rational expression equal to zero, it's equally important to identify the excluded values. These are the values of x that make the denominator zero, thus rendering the expression undefined. In our case, the denominator is (x - 4)(x + 4). To find the excluded values, we set the denominator equal to zero:

(x−4)(x+4)=0(x - 4)(x + 4) = 0

This equation is satisfied if either factor is zero. Therefore, we have two possibilities:

  1. x - 4 = 0, which gives x = 4
  2. x + 4 = 0, which gives x = -4

So, the excluded values are x = 4 and x = -4. These values are not in the domain of the rational expression because they make the denominator zero. Recognizing and excluding these values is crucial for a complete understanding of the expression's behavior.

In the context of the problem, we found that x = 9 makes the rational expression equal to zero. Since 9 is not an excluded value, it is a valid solution. However, if we had found a value that also made the denominator zero, we would have to discard it.

Understanding excluded values is particularly important when graphing rational functions. The excluded values correspond to vertical asymptotes on the graph, which are vertical lines that the function approaches but never touches. Recognizing these asymptotes helps in accurately sketching the graph of the function and understanding its behavior near these points.

In summary, always check for excluded values when working with rational expressions. This involves setting the denominator equal to zero and solving for x. These values must be excluded from the solution set and considered when analyzing the expression or its corresponding function.

Why the Numerator Matters

The numerator of a rational expression plays a pivotal role in determining when the expression equals zero. As discussed earlier, a fraction is zero if and only if its numerator is zero and its denominator is not zero. This principle is fundamental to solving many problems involving rational expressions and equations.

To understand why this is the case, consider the definition of a fraction. A fraction $\frac{a}{b}$ represents the result of dividing a by b. If a (the numerator) is zero, then we are dividing zero by b (the denominator). As long as b is not zero, the result of this division is always zero. This is a basic property of arithmetic.

However, if both the numerator and the denominator are zero, we have an indeterminate form (0/0), which is not defined. This is why we must always check that the values of x that make the numerator zero do not also make the denominator zero. If they do, those values are not valid solutions.

The numerator also provides information about the x-intercepts of the graph of a rational function. The x-intercepts are the points where the graph crosses the x-axis, which occur when the function value (y-value) is zero. Since a rational expression is zero when its numerator is zero, the zeros of the numerator correspond to the x-intercepts of the graph, provided they are not excluded values.

For instance, in our example, the numerator is x - 9, which equals zero when x = 9. This means that the graph of the rational function $\frac{x-9}{(x-4)(x+4)}$ has an x-intercept at x = 9. This connection between the numerator and the x-intercepts is a valuable tool for graphing and analyzing rational functions.

In conclusion, the numerator of a rational expression is crucial for determining when the expression is zero and for understanding the behavior of the corresponding function. By focusing on the numerator and ensuring that the denominator is not simultaneously zero, we can accurately solve equations and analyze rational expressions.

Common Mistakes to Avoid

When working with rational expressions, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you approach problems more effectively and avoid errors. Here are some of the most frequent mistakes:

  1. Forgetting to Check the Denominator:

    • One of the most common errors is finding the values of x that make the numerator zero but failing to check whether these values also make the denominator zero. As we've emphasized, if a value makes both the numerator and the denominator zero, it is not a solution. Always verify that your solutions do not lead to division by zero.

  2. Incorrectly Simplifying the Expression:

    • Simplifying rational expressions can be tricky. Make sure to only cancel factors that are common to both the numerator and the denominator. Avoid canceling terms, which are parts of a sum or difference. For example, in the expression $ rac{x-9}{(x-4)(x+4)}$, you cannot cancel the x terms because they are part of larger expressions.

  3. Ignoring Excluded Values:

    • Failing to identify and exclude the values that make the denominator zero can lead to incorrect conclusions. These excluded values are critical for understanding the domain of the rational expression and the behavior of its graph. Always determine the excluded values before solving equations or analyzing the expression.

  4. Algebraic Errors:

    • Simple algebraic mistakes, such as incorrect factoring or sign errors, can derail the entire solution process. Double-check each step of your work, especially when dealing with more complex expressions. Pay close attention to the rules of algebra and arithmetic.

  5. Misunderstanding the Question:

    • Ensure you fully understand what the problem is asking. Are you asked to find when the expression is zero, undefined, or equal to a specific value? Misinterpreting the question can lead to solving for the wrong thing. Read the problem carefully and identify the key information.

By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence when working with rational expressions. Practice and attention to detail are key to success in this area.

Conclusion

In conclusion, determining the value of x for which the rational expression $\frac{x-9}{(x-4)(x+4)}$ equals zero involves a clear and systematic approach. The key principle is that a fraction is zero if and only if its numerator is zero and its denominator is not zero. By setting the numerator (x - 9) equal to zero and solving for x, we found x = 9. We then verified that this value does not make the denominator zero, confirming that x = 9 is the solution.

Throughout this article, we've emphasized the importance of understanding rational expressions, identifying excluded values, and avoiding common mistakes. By checking the denominator and recognizing potential pitfalls, you can approach similar problems with confidence and accuracy. The step-by-step solution process provided a clear roadmap for solving equations involving rational expressions.

The concepts discussed here are fundamental to algebra and calculus, and mastering them will significantly enhance your mathematical skills. Rational expressions appear in various contexts, from solving equations to graphing functions, making a solid understanding essential for further studies in mathematics. Remember to always focus on the numerator, check the denominator, and be mindful of excluded values to ensure accurate solutions.