Solving 0 = (2x + 7) / (x^2 + 5) A Step-by-Step Guide

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Introduction

In mathematics, solving equations is a fundamental skill. This article provides a comprehensive guide to solving the equation 0 = (2x + 7) / (x^2 + 5). This equation involves a rational expression, where a polynomial is divided by another polynomial. Our goal is to find the value(s) of x that make this equation true. Understanding how to solve such equations is crucial for various mathematical applications, including calculus, algebra, and beyond. This guide will walk you through the steps, providing clear explanations and insights along the way. To effectively solve this equation, we must understand the conditions under which a fraction can equal zero and then apply algebraic techniques to isolate x. We will also delve into why certain potential solutions must be verified to ensure they are valid. By the end of this guide, you will have a solid understanding of how to approach similar problems and the reasoning behind each step.

Understanding the Problem

The given equation is 0 = (2x + 7) / (x^2 + 5). The key to solving this equation lies in recognizing that a fraction is equal to zero only if its numerator is zero and its denominator is not zero. If the denominator were zero, the fraction would be undefined, making the entire equation meaningless. Thus, our primary focus is on the numerator, 2x + 7. We need to find the value(s) of x that make 2x + 7 equal to zero. This is because any non-zero number divided by zero is undefined. To ensure a valid solution, we must also verify that the value(s) of x we find do not make the denominator, x^2 + 5, equal to zero. The denominator, x^2 + 5, presents an interesting aspect of the problem. Since x is squared, any real number substituted for x will result in a non-negative value when squared. Adding 5 to this non-negative value ensures that the denominator will always be greater than or equal to 5, never reaching zero. Therefore, we only need to concentrate on making the numerator equal to zero to solve this equation effectively. This simplifies the problem significantly, allowing us to focus on solving the linear equation in the numerator.

Step-by-Step Solution

  1. Set the numerator equal to zero: The first step is to recognize that for the fraction (2x + 7) / (x^2 + 5) to be equal to zero, the numerator (2x + 7) must be equal to zero. This is a fundamental principle when dealing with fractions and equations. We set up the equation as follows:

    2x + 7 = 0

    This equation is a linear equation, which we can solve using basic algebraic techniques. Linear equations are among the simplest types of equations to solve, making this problem relatively straightforward once we understand this initial step.

  2. Isolate x: To solve this equation, we need to isolate x on one side of the equation. We start by subtracting 7 from both sides:

    2x + 7 - 7 = 0 - 7

    This simplifies to:

    2x = -7

    Subtracting the same value from both sides maintains the equality, allowing us to progress towards isolating x. The next step involves dealing with the coefficient of x.

  3. Divide by the coefficient of x: Now, we divide both sides of the equation by 2, the coefficient of x:

    (2x) / 2 = -7 / 2

    This gives us:

    x = -7/2

    By dividing both sides by the coefficient of x, we finally isolate x and find its value. This value is a potential solution to our original equation, but we still need to verify that it doesn't make the denominator zero.

  4. Verify the solution: We found that x = -7/2. To ensure this is a valid solution, we need to check if it makes the denominator, x^2 + 5, equal to zero. Substitute x = -7/2 into the denominator:

    (-7/2)^2 + 5 = (49/4) + 5 = (49/4) + (20/4) = 69/4

    Since 69/4 is not equal to zero, the denominator is not zero when x = -7/2. This confirms that our solution is valid. If the denominator had turned out to be zero, we would have had to discard that solution because it would make the original expression undefined. However, in this case, the verification step confirms the correctness of our solution.

Checking the Solution

To ensure the solution x = -7/2 is correct, we can substitute it back into the original equation: 0 = (2x + 7) / (x^2 + 5). This step is crucial in solving this equation as it verifies our algebraic manipulations and confirms that the value of x we found indeed makes the equation true. We will meticulously perform this check to leave no room for doubt. Substituting x = -7/2 into the original equation, we get: 0 = [2(-7/2) + 7] / [(-7/2)^2 + 5]. First, let’s simplify the numerator: 2(-7/2) + 7 = -7 + 7 = 0. As the numerator is zero, the entire fraction becomes zero, provided the denominator is not zero. Next, we examine the denominator: (-7/2)^2 + 5 = (49/4) + 5. To add these terms, we convert 5 to a fraction with a denominator of 4: 5 = 20/4. So, the denominator becomes (49/4) + (20/4) = 69/4, which is not zero. Thus, we have: 0 = 0 / (69/4). Since zero divided by any non-zero number is zero, this equation holds true. This verification process solidifies our confidence in the solution x = -7/2, confirming that it satisfies the original equation. This thorough check is a best practice in mathematics, ensuring accuracy and preventing errors. Verifying the solution is particularly important in complex equations where mistakes can easily occur during the solving process.

Conclusion

In conclusion, the solution to the equation 0 = (2x + 7) / (x^2 + 5) is x = -7/2. We arrived at this solution by setting the numerator of the fraction equal to zero and solving for x. The process involved isolating x through algebraic manipulation and then verifying that the solution did not make the denominator zero. This step-by-step approach is crucial for solving equations involving rational expressions. Understanding the underlying principles, such as the condition for a fraction to be zero, is fundamental in mathematics. Additionally, the verification step ensures that our solution is valid and accurate. This methodical approach can be applied to various types of equations, making it a valuable skill in mathematics. By following these steps, one can confidently solve this equation and similar problems involving rational expressions. This comprehensive guide has illustrated the key steps and considerations in the process, offering a clear and thorough understanding of the solution. Mathematical problem-solving requires both understanding the concepts and applying the correct procedures, and this example demonstrates the importance of both.