Solving (x^2+11x)/(x-2) = -30/(x-2) A Step-by-Step Guide

In mathematics, solving equations is a fundamental skill. This article delves into the process of solving a specific type of equation: a rational equation. We will break down the steps required to find the solution set for the equation (x^2 + 11x) / (x - 2) = -30 / (x - 2). This comprehensive guide is designed to provide a clear understanding of the methods involved, ensuring you can confidently tackle similar problems in the future. Mastering the art of solving rational equations not only strengthens your algebra skills but also lays a solid foundation for more advanced mathematical concepts. So, let's embark on this journey together and unravel the intricacies of this equation.

Understanding Rational Equations

Before we dive into the solution, it's crucial to understand what a rational equation is. A rational equation is an equation that contains one or more rational expressions. A rational expression, in turn, is a fraction where the numerator and the denominator are polynomials. The equation we are dealing with, (x^2 + 11x) / (x - 2) = -30 / (x - 2), perfectly fits this definition. It's essential to recognize the structure of a rational equation because the methods used to solve them differ from those used for linear or quadratic equations. The key to solving rational equations lies in eliminating the fractions, which we will explore in detail in the following sections. Furthermore, understanding the domain of the equation is critical. The domain is the set of all possible values of x that do not make the denominator zero. This is a crucial step to avoid extraneous solutions, which are solutions that arise during the solving process but do not satisfy the original equation. Identifying the domain helps ensure that our final solution set is accurate and valid. By grasping these fundamental concepts, we set the stage for a successful and methodical approach to solving the equation.

Step 1: Identifying Restrictions

The first crucial step in solving any rational equation is to identify any restrictions on the variable. Restrictions occur when the denominator of a rational expression equals zero, as division by zero is undefined in mathematics. In our equation, (x^2 + 11x) / (x - 2) = -30 / (x - 2), the denominator is (x - 2). To find the restriction, we set the denominator equal to zero and solve for x: x - 2 = 0. Adding 2 to both sides, we find that x = 2. This means that x cannot be equal to 2, as it would make the denominator zero, rendering the equation undefined. Therefore, x = 2 is a restriction on our solution. Identifying this restriction is paramount because it helps us avoid extraneous solutions later on. An extraneous solution is a value that we obtain as a solution through the algebraic process but does not satisfy the original equation. This often happens in rational equations due to the manipulation of the equation. By acknowledging and noting the restriction early on, we can effectively filter out any extraneous solutions, ensuring that the solutions we obtain are valid. This careful attention to detail is a hallmark of accurate and reliable mathematical problem-solving. The restriction x = 2 will be pivotal in the final verification of our solution set.

Step 2: Eliminating the Denominator

The next step in solving the rational equation (x^2 + 11x) / (x - 2) = -30 / (x - 2) is to eliminate the denominator. This simplifies the equation and allows us to work with a more manageable form. Since both sides of the equation have the same denominator, (x - 2), we can multiply both sides of the equation by this denominator to eliminate it. This process is based on the fundamental principle of equality, which states that if you perform the same operation on both sides of an equation, the equality holds. Multiplying both sides by (x - 2) gives us: (x - 2) * [(x^2 + 11x) / (x - 2)] = (x - 2) * [-30 / (x - 2)]. On both sides, the (x - 2) terms cancel out, leaving us with a simplified equation: x^2 + 11x = -30. This elimination of the denominator transforms the rational equation into a quadratic equation, which is much easier to solve. It's important to remember that multiplying both sides by an expression containing a variable can sometimes introduce extraneous solutions, which is why we identified restrictions in the previous step. This step is crucial in transforming a complex rational equation into a simpler algebraic form, setting the stage for solving for x. The resulting quadratic equation is now our focus, and we will apply standard methods to find its solutions.

Step 3: Solving the Quadratic Equation

After eliminating the denominator, we arrived at the quadratic equation x^2 + 11x = -30. To solve this equation, we first need to set it equal to zero. This is achieved by adding 30 to both sides of the equation, resulting in: x^2 + 11x + 30 = 0. Now we have a standard quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 11, and c = 30. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is the most straightforward approach. We need to find two numbers that multiply to 30 and add up to 11. These numbers are 5 and 6. Therefore, we can factor the quadratic equation as follows: (x + 5)(x + 6) = 0. To find the solutions for x, we set each factor equal to zero: x + 5 = 0 and x + 6 = 0. Solving these linear equations gives us x = -5 and x = -6. These are the potential solutions to our quadratic equation. However, we must remember the restriction we identified earlier, which is x ≠ 2. Since neither of our potential solutions equals 2, they both remain as candidates for the final solution set. The process of solving the quadratic equation is a critical step in finding the values of x that satisfy the original rational equation. We now have two potential solutions, and the next step is to verify their validity.

Step 4: Verifying the Solutions

Once we have potential solutions for the equation, it is essential to verify them against the original equation and any restrictions identified earlier. Our potential solutions are x = -5 and x = -6, and the restriction we found was x ≠ 2. Since neither -5 nor -6 equals 2, they both pass the initial restriction check. Now, we need to substitute each solution back into the original equation, (x^2 + 11x) / (x - 2) = -30 / (x - 2), to ensure they satisfy the equation. Let's start with x = -5: [((-5)^2 + 11(-5)) / (-5 - 2)] = -30 / (-5 - 2) [(25 - 55) / (-7)] = -30 / (-7) [-30 / -7] = -30 / -7. This simplifies to 30/7 = 30/7, which is true. Therefore, x = -5 is a valid solution. Next, let's verify x = -6: [((-6)^2 + 11(-6)) / (-6 - 2)] = -30 / (-6 - 2) [(36 - 66) / (-8)] = -30 / (-8) [-30 / -8] = -30 / -8. This simplifies to 30/8 = 30/8, which is also true. Thus, x = -6 is also a valid solution. Verifying the solutions is a crucial step in solving rational equations. It ensures that the solutions we obtained through algebraic manipulation are indeed solutions to the original problem and not extraneous solutions. This rigorous verification process adds a layer of confidence to our final solution set. With both solutions verified, we can now confidently state the solution set for the equation.

Final Solution Set

After carefully solving the rational equation (x^2 + 11x) / (x - 2) = -30 / (x - 2), identifying restrictions, and verifying potential solutions, we have arrived at the final solution set. Our potential solutions were x = -5 and x = -6, and both of these values were confirmed to satisfy the original equation. Furthermore, neither of these values violated the restriction x ≠ 2, which we identified at the outset. Therefore, the solution set for the equation is {-5, -6}. This means that when x is either -5 or -6, the equation holds true. Presenting the solution set in this format provides a clear and concise answer to the problem. The journey of solving this rational equation has taken us through several key steps, including identifying restrictions, eliminating denominators, solving a quadratic equation, and verifying solutions. Each step is crucial in the process, and mastering these steps is fundamental to solving a wide range of mathematical problems. Understanding the solution set not only provides the answer to the specific equation but also deepens our understanding of the behavior of rational equations and their solutions. The solution set {-5, -6} is the culmination of our methodical approach and careful execution of each step.

In conclusion, solving rational equations requires a systematic approach. By carefully identifying restrictions, eliminating denominators, solving the resulting equation (in this case, a quadratic), and verifying the solutions, we can accurately determine the solution set. For the equation (x^2 + 11x) / (x - 2) = -30 / (x - 2), the solution set is {-5, -6}. This step-by-step guide not only provides the answer but also equips you with the skills to tackle similar problems confidently. Remember, each step is crucial, and attention to detail is key to avoiding errors and extraneous solutions. With practice, solving rational equations becomes a manageable and rewarding mathematical exercise. This understanding forms a vital part of your mathematical toolkit, enabling you to solve more complex problems in the future. The process of solving this equation highlights the interconnectedness of various algebraic concepts and underscores the importance of a thorough and methodical approach in mathematics.