Solving The Rational Equation X^2-5x-14 / X+2 = -2 A Step-by-Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. Among the various types of equations, rational equations, which involve fractions with polynomials in the numerator and denominator, often pose a unique challenge. This article delves into the process of solving the rational equation x2−5x−14x+2=−2\frac{x^2-5 x-14}{x+2}=-2, providing a step-by-step guide and emphasizing the importance of checking for extraneous solutions.

1. Understanding Rational Equations

Before diving into the solution, it's crucial to grasp the concept of rational equations. A rational equation is an equation that contains one or more rational expressions. A rational expression is simply a fraction where the numerator and denominator are polynomials. Solving rational equations requires a slightly different approach than solving linear or quadratic equations, primarily because we need to be mindful of potential restrictions on the variable. These restrictions arise from the denominator of the rational expression, as division by zero is undefined. Therefore, any value of the variable that makes the denominator zero must be excluded from the solution set.

The key strategy for solving rational equations is to eliminate the fractions. This is achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. By doing so, we transform the rational equation into a simpler polynomial equation that can be solved using familiar techniques. However, it is absolutely essential to check the solutions obtained against the original equation to ensure they do not make any of the denominators zero. Solutions that do are called extraneous solutions and must be discarded.

2. Solving the Equation x2−5x−14x+2=−2\frac{x^2-5 x-14}{x+2}=-2

Now, let's tackle the given equation: x2−5x−14x+2=−2\frac{x^2-5 x-14}{x+2}=-2. This equation presents a rational expression on the left-hand side and a constant on the right-hand side. Our goal is to isolate the variable x and find the values that satisfy the equation.

2.1. Identifying Restrictions

The first step in solving any rational equation is to identify any values of x that would make the denominator zero. In this case, the denominator is x + 2. Setting this equal to zero, we get x + 2 = 0, which gives us x = -2. Therefore, x cannot be equal to -2, as this would result in division by zero, making the expression undefined. We must keep this restriction in mind and check our solutions later to ensure that x = -2 is not a solution.

2.2. Eliminating the Fraction

To eliminate the fraction, we multiply both sides of the equation by the denominator, which is x + 2:

(x + 2) * (\frac{x^2-5 x-14}{x+2}) = -2 * (x + 2)

This simplifies to:

x^2 - 5x - 14 = -2(x + 2)

2.3. Simplifying and Rearranging

Next, we expand the right-hand side and rearrange the equation to obtain a quadratic equation:

x^2 - 5x - 14 = -2x - 4

Adding 2x and 4 to both sides, we get:

x^2 - 5x + 2x - 14 + 4 = 0

Simplifying further:

x^2 - 3x - 10 = 0

2.4. Solving the Quadratic Equation

We now have a quadratic equation in the standard form ax^2 + bx + c = 0. 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 -10 and add up to -3. These numbers are -5 and 2.

Therefore, we can factor the quadratic equation as follows:

(x - 5)(x + 2) = 0

Setting each factor equal to zero, we get:

x - 5 = 0 or x + 2 = 0

Solving for x, we find two potential solutions:

x = 5 or x = -2

2.5. Checking for Extraneous Solutions

As mentioned earlier, it is crucial to check for extraneous solutions. We identified x = -2 as a restriction because it makes the denominator of the original equation zero. Therefore, x = -2 is an extraneous solution and must be discarded.

Now, let's check x = 5 by substituting it back into the original equation:

52−5(5)−145+2=−2\frac{5^2-5(5)-14}{5+2} = -2

25−25−147=−2\frac{25-25-14}{7} = -2

−147=−2\frac{-14}{7} = -2

-2 = -2

Since the equation holds true for x = 5, it is a valid solution.

2.6. The Solution

Therefore, the only solution to the equation x2−5x−14x+2=−2\frac{x^2-5 x-14}{x+2}=-2 is x = 5.

3. Extraneous Solutions: A Deeper Dive

Extraneous solutions are a common pitfall in solving rational equations. They arise when the process of solving the equation introduces solutions that do not satisfy the original equation. This typically happens when we multiply both sides of the equation by an expression containing the variable, as this can inadvertently introduce solutions that make the denominator zero.

To avoid extraneous solutions, it is imperative to always check the solutions obtained by substituting them back into the original equation. If a solution makes any of the denominators zero, it is an extraneous solution and must be discarded.

Consider the simple equation xx−2=2x−2\frac{x}{x-2} = \frac{2}{x-2}. Multiplying both sides by (x - 2) gives us x = 2. However, if we substitute x = 2 back into the original equation, we get 22−2=22−2\frac{2}{2-2} = \frac{2}{2-2}, which simplifies to 20=20\frac{2}{0} = \frac{2}{0}. Since division by zero is undefined, x = 2 is an extraneous solution, and the equation has no solution.

4. Common Mistakes to Avoid

Solving rational equations can be tricky, and there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and solve equations more accurately.

  • Forgetting to Identify Restrictions: As emphasized throughout this article, identifying restrictions is the most critical step in solving rational equations. Failing to do so can lead to extraneous solutions being included in the solution set.
  • Incorrectly Eliminating Fractions: When multiplying both sides of the equation by the LCM of the denominators, ensure that you multiply every term on both sides. Missing a term can lead to an incorrect equation and ultimately, wrong solutions.
  • Not Checking for Extraneous Solutions: This is arguably the most common mistake. Even if you have followed all the steps correctly, it is still crucial to check for extraneous solutions. Simply forgetting this step can render your entire solution incorrect.
  • Incorrectly Solving the Resulting Equation: After eliminating the fractions, you will be left with a polynomial equation. Make sure you solve this equation correctly, using appropriate techniques such as factoring, completing the square, or the quadratic formula. Errors in solving this equation will lead to incorrect solutions.

5. Real-World Applications of Rational Equations

Rational equations are not just abstract mathematical concepts; they have numerous applications in the real world. They are used to model various phenomena in physics, engineering, economics, and other fields. Here are a few examples:

  • Rate Problems: Rational equations are often used to solve problems involving rates, such as work rates, speed, and time. For instance, if two people are working together on a task, the combined work rate can be expressed as a rational equation.
  • Mixture Problems: Problems involving mixtures of different concentrations can also be modeled using rational equations. For example, determining the amount of a solution needed to achieve a desired concentration involves solving a rational equation.
  • Electrical Circuits: In electrical engineering, rational equations are used to analyze circuits involving resistors, capacitors, and inductors. The impedance of a circuit, which is the opposition to the flow of alternating current, is often expressed as a rational function.
  • Lens Equation: In optics, the lens equation, which relates the focal length of a lens to the object distance and image distance, is a rational equation.

6. Conclusion

Solving rational equations requires a systematic approach, including identifying restrictions, eliminating fractions, solving the resulting equation, and most importantly, checking for extraneous solutions. By understanding the underlying principles and common pitfalls, you can master the art of solving rational equations and apply this knowledge to various real-world problems. Remember, practice is key to success in mathematics. The more you solve rational equations, the more comfortable and confident you will become. So, grab your pencil and paper, and start solving!