Solving The Equation: $-1 + 3/x = 9/x^2$ - Math Guide

Introduction

In this comprehensive guide, we will delve into the process of solving the equation $-1 + \frac{3}{x} = \frac{9}{x^2}$. This equation falls under the category of rational equations, which often require a few algebraic manipulations to simplify and ultimately solve. Understanding how to tackle such equations is crucial for anyone studying algebra or related fields. We will walk through each step in detail, ensuring clarity and providing insights along the way. This article is designed to help students, educators, and anyone interested in mathematics to grasp the fundamental techniques involved in solving rational equations. Our primary focus will be on transforming the equation into a more manageable form, identifying any restrictions on the variable x, and employing standard algebraic methods to find the solutions. So, let's embark on this mathematical journey and unravel the solution together!

Understanding the Equation and Identifying Restrictions

Before we jump into solving the equation, it's essential to understand its structure and identify any restrictions on the variable x. Our equation is $-1 + \frac{3}{x} = \frac{9}{x^2}$. Notice that x appears in the denominators of the fractions. This immediately tells us that x cannot be equal to zero, because division by zero is undefined in mathematics. This is a critical restriction that we must keep in mind as we solve the equation. Failing to account for this could lead to extraneous solutions, which are solutions that we find algebraically but do not satisfy the original equation.

Identifying restrictions is a fundamental step in solving any equation involving fractions or radicals. It ensures that our solutions are valid within the context of the equation. In this case, the restriction $x \neq 0$ is straightforward, but it's vital to note it explicitly. Now that we have this restriction in mind, we can proceed with manipulating the equation to solve for x. The next step typically involves clearing the fractions to make the equation easier to work with. This often involves multiplying both sides of the equation by a common denominator, which we'll discuss in the next section. By recognizing and addressing the restrictions early on, we pave the way for a more accurate and reliable solution process. Remember, mathematical rigor is crucial for achieving correct results.

Clearing the Fractions

To solve the equation $-1 + \frac{3}{x} = \frac{9}{x^2}$, the first step toward simplification is to eliminate the fractions. We can achieve this by multiplying both sides of the equation by the least common denominator (LCD). In this case, the LCD is $x^2$, as it is the smallest expression that both x and $x^2$ will divide into evenly. By multiplying both sides by $x^2$, we effectively clear the fractions and transform the equation into a more manageable form.

Let's walk through this process step by step. Multiplying both sides of the equation by $x^2$, we get:

$x^2 \times (-1 + \frac{3}{x}) = x^2 \times \frac{9}{x^2}$

Distributing $x^2$ on the left side gives us:

$-x^2 + 3x = 9$

Now, we have a quadratic equation, which is much easier to handle than the original rational equation. This transformation is a key technique in solving equations involving fractions. Clearing the fractions allows us to work with a polynomial equation, for which we have well-established methods for finding solutions. It's important to perform this step carefully, ensuring that the multiplication is correctly distributed across all terms. Once the fractions are cleared, the equation becomes significantly less daunting, and we can proceed with standard algebraic techniques to find the values of x that satisfy the equation. The next step is to rearrange the quadratic equation into standard form, which we'll cover in the following section. Remember, accuracy in each step is crucial for reaching the correct solution.

Rearranging into Standard Quadratic Form

Now that we've cleared the fractions, our equation is $-x^2 + 3x = 9$. To solve this, we need to rearrange it into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. This form allows us to easily apply various methods, such as factoring, completing the square, or using the quadratic formula, to find the solutions for x. The standard form is a crucial stepping stone in solving quadratic equations, and mastering this step is essential.

To rearrange our equation, we'll add $x^2$ and subtract $3x$ from both sides, and subtract 9 from both sides, resulting in:

$0 = x^2 - 3x + 9$

So, our equation in standard form is $x^2 - 3x + 9 = 0$. Here, we can identify the coefficients: $a = 1$, $b = -3$, and $c = 9$. These coefficients will be particularly useful if we choose to use the quadratic formula to solve the equation.

Rearranging the equation into standard form is a fundamental step because it sets the stage for applying a variety of solution techniques. Without this step, it would be significantly more challenging to find the solutions. Understanding this process is key to solving not only this specific equation but also a wide range of quadratic equations. Now that we have the equation in standard form, we can move on to selecting and applying a method to find the roots. In the next section, we'll explore using the quadratic formula, which is a reliable method for solving any quadratic equation, regardless of whether it can be easily factored. Remember, the ability to rearrange equations is a cornerstone of algebraic manipulation.

Applying the Quadratic Formula

With our equation in the standard quadratic form $x^2 - 3x + 9 = 0$, we can now apply the quadratic formula to find the solutions for x. The quadratic formula is a powerful tool that provides the solutions for any quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our equation, we have $a = 1$, $b = -3$, and $c = 9$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(9)}}{2(1)}$

Simplifying further:

$x = \frac{3 \pm \sqrt{9 - 36}}{2}$

$x = \frac{3 \pm \sqrt{-27}}{2}$

Notice that we have a negative number under the square root, which means that the solutions will be complex numbers. This is perfectly acceptable and indicates that the parabola represented by the quadratic equation does not intersect the x-axis.

The quadratic formula is a cornerstone of algebra, providing a reliable method for solving quadratic equations. Understanding how to apply it correctly is crucial for success in mathematics. The fact that we obtained complex solutions tells us valuable information about the nature of the roots and the graph of the equation. Continuing with the simplification, we can express the square root of -27 as:

$\sqrt{-27} = \sqrt{27} \times \sqrt{-1} = 3\sqrt{3}i$

Where i is the imaginary unit, defined as $i = \sqrt{-1}$.

Substituting this back into our equation, we get:

$x = \frac{3 \pm 3\sqrt{3}i}{2}$

So, the solutions are:

$x = \frac{3 + 3\sqrt{3}i}{2}$ and $x = \frac{3 - 3\sqrt{3}i}{2}$

These are the complex solutions to our original equation. It's important to express the solutions in their simplest form and to recognize the presence of complex numbers when they arise. In the next section, we'll summarize our solutions and ensure they align with any restrictions we identified earlier. Remember, attention to detail is key when working with complex numbers.

Checking for Extraneous Solutions and Summarizing the Solutions

After finding the solutions to an equation, it's crucial to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original equation. This is particularly important when dealing with rational equations, where we have restrictions on the variable. Earlier, we identified that x cannot be equal to 0 for our equation $-1 + \frac{3}{x} = \frac{9}{x^2}$.

Our solutions are $x = \frac{3 + 3\sqrt{3}i}{2}$ and $x = \frac{3 - 3\sqrt{3}i}{2}$. Both of these solutions are complex numbers and are clearly not equal to 0. Therefore, they do not violate our restriction, and we can confidently say that they are valid solutions to the original equation.

Checking for extraneous solutions is a vital step in the problem-solving process. It ensures that we only accept solutions that truly satisfy the original equation. This step is often overlooked, but it can be the difference between a correct and an incorrect answer.

Now, let's summarize our solutions. The equation $-1 + \frac{3}{x} = \frac{9}{x^2}$ has two complex solutions:

$x = \frac{3 + 3\sqrt{3}i}{2}$ and $x = \frac{3 - 3\sqrt{3}i}{2}$

These solutions represent the values of x that make the equation true. We have successfully navigated through the steps of clearing fractions, rearranging into standard quadratic form, applying the quadratic formula, and checking for extraneous solutions. This process demonstrates a comprehensive approach to solving rational equations. Remember, a systematic approach is often the key to success in mathematics.

Conclusion

In this guide, we've thoroughly explored the process of solving the equation $-1 + \frac{3}{x} = \frac{9}{x^2}$. We started by understanding the equation and identifying the restriction $x \neq 0$. Then, we cleared the fractions by multiplying both sides by $x^2$, which transformed the equation into a quadratic form. We rearranged the quadratic equation into the standard form $x^2 - 3x + 9 = 0$ and applied the quadratic formula to find the solutions. Understanding these steps is crucial for solving a wide range of mathematical problems.

The application of the quadratic formula led us to complex solutions: $x = \frac{3 + 3\sqrt{3}i}{2}$ and $x = \frac{3 - 3\sqrt{3}i}{2}$. We verified that these solutions are not extraneous by confirming that they do not violate our initial restriction. This comprehensive approach underscores the importance of not only finding solutions but also ensuring their validity within the context of the original equation.

Mastering the techniques discussed here—clearing fractions, rearranging equations, applying the quadratic formula, and checking for extraneous solutions—is essential for anyone studying algebra or related fields. These skills are not only applicable to specific equations like the one we've solved but also form the foundation for more advanced mathematical concepts. By diligently practicing these methods, you can enhance your problem-solving abilities and gain confidence in your mathematical skills. Remember to always check your solutions and ensure they make sense in the original context of the problem. For further learning and exploration of mathematical concepts, consider visiting resources like Khan Academy, which offers a wealth of educational materials and exercises.