Solving Rational Functions A Step By Step Guide To F(x) = (5x + 14) / (6x^2 + X - 2) = 1
Introduction to Rational Functions
Rational functions are a fundamental concept in algebra and calculus, playing a crucial role in various mathematical and real-world applications. Understanding how to solve equations involving rational functions is an essential skill for anyone delving into higher mathematics or fields like engineering, physics, and economics. In this article, we will explore a specific problem: solving the rational function f(x) = (5x + 14) / (6x^2 + x - 2) = 1. We'll break down the steps, explain the underlying principles, and provide insights to help you tackle similar problems with confidence. Our main focus will be on providing a thorough and easy-to-understand guide that caters to both beginners and those looking to solidify their understanding.
To begin, it’s important to define what a rational function is. A rational function is any function that can be expressed as the quotient of two polynomials. In other words, it's a function of the form f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions, and Q(x) is not equal to zero. The function we are examining, f(x) = (5x + 14) / (6x^2 + x - 2), perfectly fits this definition, making it a classic example of a rational function. The complexity often arises from the denominator, which, in this case, is a quadratic expression. Solving such functions involves manipulating algebraic expressions, factoring polynomials, and ensuring we account for any restrictions on the domain.
When solving rational equations, one of the primary goals is to find the values of x that make the equation true. This often involves setting the rational function equal to a specific value, in our case, 1. The process typically includes clearing the fractions by multiplying both sides of the equation by the denominator, which transforms the rational equation into a polynomial equation. Once we have a polynomial equation, we can use various techniques such as factoring, the quadratic formula, or other algebraic methods to find the solutions. However, it’s absolutely critical to check these solutions against the original equation to ensure they are valid. This is because multiplying by the denominator can introduce extraneous solutions – solutions that satisfy the transformed polynomial equation but not the original rational equation. Extraneous solutions occur when a solution makes the original denominator equal to zero, which is undefined for a rational function. Therefore, a rigorous verification step is a non-negotiable part of solving rational equations. This meticulous approach will ensure accurate and reliable results.
Problem Statement: f(x) = (5x + 14) / (6x^2 + x - 2) = 1
Our specific problem is to solve the equation f(x) = (5x + 14) / (6x^2 + x - 2) = 1. This means we need to find all values of x that satisfy this equation. To achieve this, we will follow a step-by-step approach: first, we will clear the fraction by multiplying both sides of the equation by the denominator. This will transform the rational equation into a quadratic equation. Next, we will solve the quadratic equation using factoring or the quadratic formula. Finally, we will check our solutions to ensure they are valid and not extraneous. The goal is to find all real values of x that make the given equation true, providing a complete and accurate solution.
Before diving into the algebraic manipulations, it's beneficial to understand the domain of the function. The domain of a rational function consists of all real numbers except for those values that make the denominator equal to zero. To find these excluded values, we need to determine the roots of the quadratic expression in the denominator, 6x^2 + x - 2. Setting 6x^2 + x - 2 = 0 and solving for x will give us the values that are not in the domain. This initial step is crucial because it helps us identify potential extraneous solutions later on. By knowing the restrictions on x beforehand, we can efficiently filter out invalid solutions and focus on the true solutions of the equation. The domain restrictions act as a safety net, ensuring that our final answers are mathematically sound.
In the context of solving rational equations, understanding and checking for extraneous solutions is a pivotal step. Extraneous solutions are values obtained during the solution process that do not satisfy the original equation. They typically arise when we multiply both sides of an equation by an expression that can be zero. In our problem, multiplying both sides by the denominator (6x^2 + x - 2) transforms the rational equation into a simpler polynomial equation. However, if the solutions to the polynomial equation make the denominator zero, they are extraneous because division by zero is undefined. Thus, it's absolutely necessary to substitute each potential solution back into the original rational equation to verify its validity. This process ensures that we only accept solutions that make the original equation true, preserving the integrity of our mathematical reasoning. Overlooking this step can lead to incorrect answers, highlighting the importance of a thorough verification process in solving rational functions.
Step-by-Step Solution
Step 1: Clear the Fraction
To begin solving the equation (5x + 14) / (6x^2 + x - 2) = 1, the first step is to clear the fraction. We achieve this by multiplying both sides of the equation by the denominator, which is 6x^2 + x - 2. This process eliminates the fraction and transforms the equation into a more manageable form. Multiplying both sides by (6x^2 + x - 2) gives us:
(5x + 14) = 1 * (6x^2 + x - 2)
This simplifies to:
5x + 14 = 6x^2 + x - 2
This step is critical as it converts the rational equation into a quadratic equation, which we can solve using standard algebraic techniques. Clearing the fraction is a fundamental strategy in solving rational equations because it removes the complexity of dealing with fractions and sets the stage for solving the resulting polynomial equation.
Step 2: Rearrange into a Quadratic Equation
Now that we have 5x + 14 = 6x^2 + x - 2, the next step is to rearrange the equation into the standard form of a quadratic equation, which is ax^2 + bx + c = 0. To do this, we need to move all terms to one side of the equation, setting the other side to zero. Subtracting 5x and 14 from both sides, we get:
0 = 6x^2 + x - 2 - 5x - 14
Combining like terms, we simplify the equation to:
6x^2 - 4x - 16 = 0
This quadratic equation is now in the standard form, making it easier to solve using factoring, the quadratic formula, or other methods. The correct arrangement of the equation is crucial because it allows us to directly apply standard solution techniques for quadratic equations. This step sets the foundation for the next phase of the solution process, where we will determine the values of x that satisfy the equation.
Step 3: Simplify the Quadratic Equation
Before proceeding to solve the quadratic equation 6x^2 - 4x - 16 = 0, it’s often beneficial to simplify it if possible. This can make the subsequent steps, such as factoring or applying the quadratic formula, easier and less prone to errors. In this case, we notice that all the coefficients (6, -4, and -16) are divisible by 2. Dividing the entire equation by 2 simplifies it to:
3x^2 - 2x - 8 = 0
This simplified quadratic equation is equivalent to the original but has smaller coefficients, making it easier to work with. Simplifying the equation in this way doesn't change the solutions but streamlines the solution process. This step is an example of good mathematical practice, where simplification is used to reduce complexity and improve accuracy.
Step 4: Solve the Quadratic Equation
Now that we have the simplified quadratic equation 3x^2 - 2x - 8 = 0, we need to 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 a straightforward approach. We look for two numbers that multiply to (3 * -8) = -24 and add up to -2. These numbers are -6 and 4. We can rewrite the middle term using these numbers:
3x^2 - 6x + 4x - 8 = 0
Next, we factor by grouping:
3x(x - 2) + 4(x - 2) = 0
Now, we can factor out the common term (x - 2):
(3x + 4)(x - 2) = 0
Setting each factor equal to zero gives us the potential solutions:
3x + 4 = 0 or x - 2 = 0
Solving for x, we get:
x = -4/3 or x = 2
These are the potential solutions to the quadratic equation. However, it is crucial to remember that we still need to check these solutions in the original rational equation to ensure they are not extraneous.
Step 5: Check for Extraneous Solutions
As discussed earlier, checking for extraneous solutions is a critical step in solving rational equations. We must substitute each potential solution back into the original equation, f(x) = (5x + 14) / (6x^2 + x - 2) = 1, to ensure it makes the equation true and that the denominator is not zero.
First, let’s check x = -4/3:
f(-4/3) = (5(-4/3) + 14) / (6(-4/3)^2 + (-4/3) - 2)
Simplify the numerator:
5(-4/3) + 14 = -20/3 + 14 = -20/3 + 42/3 = 22/3
Simplify the denominator:
6(-4/3)^2 + (-4/3) - 2 = 6(16/9) - 4/3 - 2 = 32/3 - 4/3 - 6/3 = 22/3
Now, divide the numerator by the denominator:
(22/3) / (22/3) = 1
So, x = -4/3 is a valid solution.
Next, let’s check x = 2:
f(2) = (5(2) + 14) / (6(2)^2 + 2 - 2)
Simplify the numerator:
5(2) + 14 = 10 + 14 = 24
Simplify the denominator:
6(2)^2 + 2 - 2 = 6(4) + 2 - 2 = 24
Now, divide the numerator by the denominator:
24 / 24 = 1
So, x = 2 is also a valid solution.
Both potential solutions satisfy the original equation, and neither makes the denominator zero. Therefore, both are valid solutions.
Final Answer
After completing all the steps, we have found two solutions for the equation f(x) = (5x + 14) / (6x^2 + x - 2) = 1. We cleared the fraction, rearranged the equation into a quadratic form, simplified it, solved the quadratic equation, and checked for extraneous solutions. The solutions we found are:
x = -4/3 and x = 2
These are the final and verified solutions to the problem. By systematically following the steps and diligently checking our work, we have ensured the accuracy and completeness of our answer. This comprehensive approach is essential for solving rational equations and handling the potential pitfalls of extraneous solutions.
Conclusion
In conclusion, solving rational equations such as f(x) = (5x + 14) / (6x^2 + x - 2) = 1 involves a systematic approach. The key steps include clearing the fraction, rearranging the equation into a standard form, solving the resulting polynomial equation, and, most importantly, checking for extraneous solutions. By following these steps diligently, we can confidently find the correct solutions and avoid common pitfalls. The process we’ve outlined provides a robust framework for tackling similar problems involving rational functions.
Understanding rational functions and their solutions is crucial in many areas of mathematics and its applications. From calculus to engineering, the ability to manipulate and solve rational equations is a valuable skill. The problem we addressed in this article serves as an excellent example of the complexities and nuances involved in working with rational functions. The methodical approach we used, with its emphasis on verification, ensures accurate and reliable results. Whether you are a student learning these concepts for the first time or a professional using them in your field, a solid grasp of these techniques is indispensable.
The importance of checking for extraneous solutions cannot be overstated. This step is often the difference between a correct solution and an incorrect one. Extraneous solutions arise because multiplying both sides of an equation by an expression that can be zero can introduce solutions that do not satisfy the original equation. This is a common issue in rational and radical equations, and a thorough check is always necessary to ensure the validity of the solutions. By making this a standard part of your problem-solving routine, you can avoid errors and build confidence in your mathematical abilities. Remember, mathematics is not just about finding answers; it’s about ensuring those answers are correct and justified. This critical mindset is what separates good problem solvers from excellent ones.
