Solving F(x) = G(x) Find X For F(x) = X/(x-2) + 1/(x-9) And G(x) = (-25x - 74)/(x^2 - 11x + 18)

In this article, we delve into the problem of finding the values of x for which two functions, f(x) and g(x), are equal. Specifically, we are given the functions f(x) = x/(x-2) + 1/(x-9) and g(x) = (-25x - 74)/(x^2 - 11x + 18). Our goal is to solve the equation f(x) = g(x). This involves algebraic manipulation, simplification, and the solution of polynomial equations. This article provides a step-by-step guide to solving this equation, offering insights into the underlying mathematical principles and techniques.

Understanding the Functions

Before diving into the solution, let's first understand the nature of the functions we are dealing with. The function f(x) is a sum of two rational functions, each with a linear denominator. Specifically, f(x) is defined as f(x) = x/(x-2) + 1/(x-9). Notice that this function has potential discontinuities at x = 2 and x = 9, as these values would make the denominators zero. The function g(x) is also a rational function, but this time the denominator is a quadratic expression. g(x) is defined as g(x) = (-25x - 74)/(x^2 - 11x + 18). We can factor the quadratic denominator as (x - 2)(x - 9), which reveals that g(x) also has potential discontinuities at x = 2 and x = 9. The fact that both functions share the same points of discontinuity suggests a potential simplification when we set them equal to each other. Understanding these potential discontinuities is crucial because they represent values of x for which the functions are not defined, and thus cannot be solutions to the equation f(x) = g(x). This initial analysis sets the stage for the algebraic manipulations that will follow, allowing us to solve for x while being mindful of the domain restrictions imposed by the denominators.

Setting Up the Equation

The initial step in solving the equation f(x) = g(x) is to explicitly set the two functions equal to each other. This means we are starting with the equation x/(x-2) + 1/(x-9) = (-25x - 74)/(x^2 - 11x + 18). This equation represents the core of our problem, and it's a rational equation because it involves fractions with polynomials in the numerator and denominator. The key to solving such equations is to eliminate the fractions by finding a common denominator. Observing the denominators, we have (x - 2), (x - 9), and (x^2 - 11x + 18). As noted earlier, the quadratic denominator can be factored as (x - 2)(x - 9). This factorization is crucial because it reveals that (x - 2)(x - 9) is indeed the common denominator for all terms in the equation. By recognizing this common denominator, we can proceed to multiply both sides of the equation by it, which will clear the fractions and transform the equation into a more manageable form. This step is pivotal in simplifying the problem and paving the way for further algebraic manipulations. Correctly identifying and utilizing the common denominator is a fundamental technique in solving rational equations, and it is a critical step in our journey to find the values of x that satisfy the given condition.

Eliminating the Fractions

To eliminate the fractions in the equation x/(x-2) + 1/(x-9) = (-25x - 74)/(x^2 - 11x + 18), we multiply both sides by the common denominator, which we identified as (x - 2)(x - 9). This crucial step clears the fractions and simplifies the equation into a more manageable polynomial equation. Multiplying both sides by (x - 2)(x - 9), we get:

(x - 2)(x - 9) * [x/(x-2) + 1/(x-9)] = (x - 2)(x - 9) * [(-25x - 74)/(x^2 - 11x + 18)]

On the left side, we distribute (x - 2)(x - 9) to both terms:

(x - 2)(x - 9) * [x/(x-2)] + (x - 2)(x - 9) * [1/(x-9)] = (x - 2)(x - 9) * [(-25x - 74)/(x^2 - 11x + 18)]

Now, we can cancel out the common factors. In the first term, (x - 2) cancels out, and in the second term, (x - 9) cancels out. On the right side, (x^2 - 11x + 18) is equivalent to (x - 2)(x - 9), so they cancel out completely. This simplification leads to:

x(x - 9) + (x - 2) = -25x - 74

This equation is now free of fractions, making it easier to work with. This step is a cornerstone of solving rational equations, as it transforms a complex equation into a simpler form that can be solved using standard algebraic techniques. The elimination of fractions allows us to focus on the polynomial terms and their relationships, bringing us closer to finding the solution for x.

Simplifying and Rearranging

Having eliminated the fractions, our equation now stands as x(x - 9) + (x - 2) = -25x - 74. The next step involves expanding and simplifying this equation to bring it into a standard polynomial form. First, we distribute the x in the first term: x^2 - 9x + (x - 2) = -25x - 74. Next, we combine like terms on the left side: x^2 - 9x + x - 2 = -25x - 74, which simplifies to x^2 - 8x - 2 = -25x - 74. Now, to solve for x, we need to rearrange the equation so that all terms are on one side, setting the equation equal to zero. This is achieved by adding 25x and 74 to both sides of the equation: x^2 - 8x - 2 + 25x + 74 = 0. Combining like terms once again, we get the quadratic equation: x^2 + 17x + 72 = 0. This simplified quadratic equation is now in the standard form ax^2 + bx + c = 0, where a = 1, b = 17, and c = 72. This form is crucial because it allows us to apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. The simplification and rearrangement process has transformed the original rational equation into a familiar and solvable quadratic equation, marking a significant step forward in our quest to find the values of x that satisfy the given conditions.

Solving the Quadratic Equation

Now that we have the quadratic equation x^2 + 17x + 72 = 0, we can solve for x. There are several methods to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. In this case, factoring is the most straightforward approach. We are looking for two numbers that multiply to 72 and add up to 17. These numbers are 8 and 9, since 8 * 9 = 72 and 8 + 9 = 17. Therefore, we can factor the quadratic equation as: (x + 8)(x + 9) = 0. Now, we apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible solutions: x + 8 = 0 or x + 9 = 0. Solving for x in each case, we get: x = -8 and x = -9. These are the potential solutions to the original equation. However, it's crucial to check these solutions against the original equation to ensure they are valid and do not result in any undefined terms (such as division by zero). This verification step is essential to ensure the accuracy and validity of our solutions. Factoring the quadratic equation has provided us with two potential solutions, bringing us closer to the final answer.

Verification of Solutions

After finding potential solutions to an equation, it's crucial to verify them by substituting them back into the original equation. This step ensures that the solutions are valid and do not lead to any undefined expressions, such as division by zero. Our potential solutions are x = -8 and x = -9. Recall the original equation: x/(x-2) + 1/(x-9) = (-25x - 74)/(x^2 - 11x + 18). Let's first verify x = -8. Substituting x = -8 into the equation, we get: (-8)/(-8-2) + 1/(-8-9) = (-25(-8) - 74)/((-8)^2 - 11(-8) + 18). Simplifying the left side: (-8)/(-10) + 1/(-17) = 4/5 - 1/17. Finding a common denominator and combining the fractions, we get: (4 * 17 - 5)/(5 * 17) = (68 - 5)/85 = 63/85. Now, let's simplify the right side: (-25(-8) - 74)/((-8)^2 - 11(-8) + 18) = (200 - 74)/(64 + 88 + 18) = 126/170. Simplifying the fraction, we get: 126/170 = 63/85. Since both sides are equal, x = -8 is a valid solution. Next, let's verify x = -9. Substituting x = -9 into the equation, we get: (-9)/(-9-2) + 1/(-9-9) = (-25(-9) - 74)/((-9)^2 - 11(-9) + 18). Simplifying the left side: (-9)/(-11) + 1/(-18) = 9/11 - 1/18. Finding a common denominator and combining the fractions, we get: (9 * 18 - 1 * 11)/(11 * 18) = (162 - 11)/198 = 151/198. Now, let's simplify the right side: (-25(-9) - 74)/((-9)^2 - 11(-9) + 18) = (225 - 74)/(81 + 99 + 18) = 151/198. Since both sides are equal, x = -9 is also a valid solution. Therefore, both x = -8 and x = -9 are solutions to the original equation. This verification step is a crucial part of the problem-solving process, ensuring that our solutions are not extraneous and that they satisfy the initial equation.

Final Answer

After solving the equation f(x) = g(x), where f(x) = x/(x-2) + 1/(x-9) and g(x) = (-25x - 74)/(x^2 - 11x + 18), and verifying the solutions, we have found that the values of x that satisfy the equation are x = -8 and x = -9. These solutions were obtained by first setting the two functions equal to each other, eliminating the fractions by multiplying by the common denominator, simplifying the resulting equation, and then solving the quadratic equation that emerged. We employed factoring as the method to solve the quadratic equation, which led us to the potential solutions. Finally, we verified these solutions by substituting them back into the original equation to ensure their validity. This comprehensive process, from the initial equation setup to the final verification, demonstrates a systematic approach to solving rational equations. The solutions x = -8 and x = -9 represent the values for which the functions f(x) and g(x) have the same output, thus providing a complete and accurate answer to the problem.