Solving 6/x² - 40/x = 14 Step-by-Step Solution Guide

This article provides a detailed, step-by-step solution to the equation 6/x² - 40/x = 14, a common type of problem encountered in algebra. We'll break down the process, explaining each step thoroughly to ensure you understand not only how to solve it but also why each step is necessary. This equation is a rational equation that can be transformed into a quadratic equation, which we can then solve using various methods. Understanding how to solve such equations is crucial for success in higher-level mathematics and related fields. So, let's dive in and conquer this equation together!

Understanding the Problem: Rational Equations and Quadratic Forms

Before we jump into the solution, let's first understand the type of equation we're dealing with. The equation 6/x² - 40/x = 14 is a rational equation. This means it involves fractions where the variable, 'x' in this case, appears in the denominator. Solving rational equations often involves eliminating the fractions to obtain a more manageable form, typically a polynomial equation. In this specific instance, we'll transform the given rational equation into a quadratic equation, which is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Recognizing this connection is the first key step in solving the problem. Quadratic equations are fundamental in algebra and have well-established methods for finding solutions, such as factoring, completing the square, or using the quadratic formula. The process of converting the rational equation to a quadratic form will involve multiplying both sides of the equation by a suitable expression to clear the denominators. This step is crucial because it allows us to apply the familiar techniques for solving quadratic equations. So, by understanding the nature of the equation and the steps involved in transforming it, we are well-prepared to tackle the solution process.

Step 1: Eliminating the Fractions – Multiplying by the Least Common Denominator

The first crucial step in solving the equation 6/x² - 40/x = 14 is to eliminate the fractions. To do this, we need to multiply both sides of the equation by the least common denominator (LCD) of the fractions present. In this case, the denominators are x² and x. The LCD is the smallest expression that is divisible by both denominators, which in this instance is x². Multiplying both sides of the equation by x² will clear the fractions and transform the equation into a more manageable form. This is a fundamental technique in solving rational equations, as it allows us to work with a polynomial equation instead of fractions. When we multiply the left side of the equation, x² * (6/x² - 40/x), we distribute the x² to each term. This gives us (x² * 6/x²) - (x² * 40/x). Simplifying these terms, we get 6 - 40x. On the right side of the equation, we have 14 * x², which gives us 14x². Therefore, multiplying both sides of the original equation by x² results in the new equation 6 - 40x = 14x². This transformed equation is a quadratic equation, which we can now solve using standard methods.

Step 2: Rearranging into Standard Quadratic Form

Now that we've eliminated the fractions and have the equation 6 - 40x = 14x², our next step is to rearrange it into the standard form of a quadratic equation, which is ax² + bx + c = 0. This standard form is crucial because it allows us to easily identify the coefficients a, b, and c, which are necessary for applying methods like the quadratic formula or factoring. To rearrange the equation, we need to move all terms to one side, leaving zero on the other side. A common approach is to move all terms to the side where the x² term has a positive coefficient. In our case, the x² term is 14x², which is already positive on the right side of the equation. Therefore, we'll move the terms on the left side to the right side. To do this, we subtract 6 from both sides and add 40x to both sides. This gives us 0 = 14x² + 40x - 6. It's often preferable to write the equation with the zero on the right side, so we can rewrite it as 14x² + 40x - 6 = 0. This is now in the standard quadratic form, with a = 14, b = 40, and c = -6. Having the equation in this form sets us up for the next step, which involves simplifying the equation by dividing by a common factor.

Step 3: Simplifying the Quadratic Equation

Before we proceed with solving the quadratic equation 14x² + 40x - 6 = 0, it's often beneficial to simplify it by dividing out any common factors from the coefficients. This makes the numbers smaller and easier to work with, especially if we plan to factor the equation. Looking at the coefficients 14, 40, and -6, we can see that they all share a common factor of 2. Dividing the entire equation by 2 means dividing each term by 2. This gives us (14x²/2) + (40x/2) - (6/2) = 0/2, which simplifies to 7x² + 20x - 3 = 0. This simplified quadratic equation is equivalent to the original but has smaller coefficients, making it easier to solve. Now we have a = 7, b = 20, and c = -3. Simplifying the equation in this way doesn't change the solutions, but it reduces the complexity of the calculations involved in the subsequent steps. With the simplified equation in hand, we can now move on to the core of the problem: solving the quadratic equation to find the values of x that satisfy it. The next step involves choosing a method for solving the quadratic equation, such as factoring, using the quadratic formula, or completing the square.

Step 4: Solving the Quadratic Equation – Factoring

Now that we have the simplified quadratic equation 7x² + 20x - 3 = 0, we can proceed to solve it. One common method for solving quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two binomials. If we can successfully factor the quadratic, we can then set each factor equal to zero and solve for x. To factor the quadratic 7x² + 20x - 3, we need to find two binomials of the form (px + q)(rx + s) such that their product equals 7x² + 20x - 3. This process often involves trial and error, but there are strategies to help. We look for two numbers that multiply to give ac (7 * -3 = -21) and add up to b (20). The numbers 21 and -1 satisfy these conditions (21 * -1 = -21 and 21 + (-1) = 20). We can then rewrite the middle term (20x) as the sum of 21x and -1x: 7x² + 21x - 1x - 3 = 0. Next, we factor by grouping: 7x(x + 3) - 1(x + 3) = 0. Notice that (x + 3) is a common factor, so we can factor it out: (7x - 1)(x + 3) = 0. Now that we have factored the quadratic equation, we can set each factor equal to zero and solve for x. This is based on 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. In the next step, we'll apply this property to find the solutions for x.

Step 5: Finding the Solutions for x

In the previous step, we successfully factored the quadratic equation 7x² + 20x - 3 = 0 into the form (7x - 1)(x + 3) = 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 means that either 7x - 1 = 0 or x + 3 = 0 (or both). Let's solve each of these linear equations separately. For the first equation, 7x - 1 = 0, we add 1 to both sides to isolate the term with x: 7x = 1. Then, we divide both sides by 7 to solve for x: x = 1/7. So, one solution is x = 1/7. For the second equation, x + 3 = 0, we subtract 3 from both sides to solve for x: x = -3. So, the other solution is x = -3. Therefore, the solutions to the quadratic equation 7x² + 20x - 3 = 0 are x = 1/7 and x = -3. These are the values of x that make the equation true. In the next step, we will verify these solutions to ensure that they are correct and that they do not lead to any undefined expressions in the original equation.

Step 6: Verifying the Solutions

After finding the solutions x = 1/7 and x = -3, it's crucial to verify them by substituting them back into the original equation: 6/x² - 40/x = 14. This step is essential to ensure that our solutions are correct and that they don't lead to any undefined expressions, such as division by zero. Let's first verify x = 1/7. Substituting x = 1/7 into the original equation, we get: 6/(1/7)² - 40/(1/7) = 14. Simplifying, we have 6/(1/49) - 40 * 7 = 14. This becomes 6 * 49 - 280 = 14, which simplifies to 294 - 280 = 14. Indeed, 14 = 14, so x = 1/7 is a valid solution. Now, let's verify x = -3. Substituting x = -3 into the original equation, we get: 6/(-3)² - 40/(-3) = 14. Simplifying, we have 6/9 + 40/3 = 14. This becomes 2/3 + 40/3 = 14, which simplifies to 42/3 = 14. Again, 14 = 14, so x = -3 is also a valid solution. Since both solutions satisfy the original equation and don't lead to any undefined expressions, we can confidently conclude that they are the correct solutions to the problem. This verification step is a critical part of the problem-solving process, as it ensures the accuracy of our results.

Step 7: Stating the Solution Set

Having verified that x = 1/7 and x = -3 are valid solutions to the equation 6/x² - 40/x = 14, we can now state the solution set. The solution set is the set of all values of x that satisfy the equation. In this case, there are two values: 1/7 and -3. We typically write the solution set using set notation, which involves enclosing the solutions within curly braces { }. Therefore, the solution set for the equation 6/x² - 40/x = 14 is {-3, 1/7}. It's important to present the solutions clearly and concisely, and set notation provides a standard way to do this. This final step concludes the problem-solving process, where we have systematically transformed the original rational equation into a quadratic equation, solved it, verified the solutions, and now stated the solution set. This comprehensive approach ensures a complete and accurate solution to the problem.

In conclusion, we have successfully solved the equation 6/x² - 40/x = 14 by following a series of steps. We began by recognizing the equation as a rational equation and transformed it into a quadratic equation by multiplying both sides by the least common denominator. We then rearranged the equation into standard quadratic form, simplified it by dividing out a common factor, and solved it by factoring. Finally, we verified our solutions and stated the solution set as {-3, 1/7}. This step-by-step process demonstrates the importance of understanding the underlying principles of algebra and applying appropriate techniques to solve equations. Solving rational and quadratic equations is a fundamental skill in mathematics, and the methods used here can be applied to a wide range of similar problems. By mastering these techniques, you can build a strong foundation for further studies in mathematics and related fields. Remember to always verify your solutions to ensure accuracy and to develop a deeper understanding of the problem-solving process.