Solving The Equation 8 + 8/x = X + 10 A Step-by-Step Guide

Jul 23, 2025 by ADMIN 59 views

Solve the Equation: A Comprehensive Guide

Are you grappling with the equation $8 + \frac{8}{x} = x + 10$ ? Don't worry, guys! You've come to the right place. This comprehensive guide will walk you through the step-by-step process of solving this equation, ensuring you not only arrive at the correct answer but also understand the underlying concepts. We'll break down each step with clear explanations and helpful tips, making it easy for you to follow along and master this type of problem. Whether you're a student tackling algebra homework or just someone looking to brush up on their math skills, this article is designed to provide you with the knowledge and confidence you need. So, let's dive in and conquer this equation together!

Understanding the Equation

Before we jump into solving, let's take a moment to understand the equation $8 + \frac{8}{x} = x + 10$ . This is a rational equation, which means it involves fractions with variables in the denominator. The presence of the fraction $\frac{8}{x}$ is what makes it a rational equation. Our goal is to find the value(s) of $x$ that make this equation true. To do this, we'll need to manipulate the equation using algebraic principles to isolate $x$ on one side. This process will involve clearing the fraction, simplifying the equation, and potentially solving a quadratic equation. Understanding the nature of the equation is the first step towards finding the solution. We need to be mindful of the domain of $x$ , as $x$ cannot be zero because division by zero is undefined. This is a crucial consideration as we proceed with solving the equation. Let's keep this in mind as we go through each step.

Step 1: Eliminate the Fraction

The first step in solving this equation is to eliminate the fraction. Fractions can make equations look intimidating, but we can easily get rid of them by multiplying both sides of the equation by the denominator. In this case, the denominator is $x$ . So, we multiply both sides of the equation $8 + \frac{8}{x} = x + 10$ by $x$ . This gives us:

$x(8 + \frac{8}{x}) = x(x + 10)$

Now, we distribute $x$ on both sides. On the left side, we have $x * 8 + x * \frac{8}{x}$ , which simplifies to $8x + 8$ . On the right side, we have $x * x + x * 10$ , which simplifies to $x^2 + 10x$ . So, our equation now looks like this:

$8x + 8 = x^2 + 10x$

By multiplying through by $x$ , we've transformed the rational equation into a quadratic equation, which is much easier to handle. This is a common technique in solving rational equations, and it sets the stage for the next steps in our solution.

Step 2: Rearrange into Quadratic Form

Now that we've eliminated the fraction, we have a new equation: $8x + 8 = x^2 + 10x$ . To solve this equation, we need to rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$ . This form allows us to use various methods for solving quadratic equations, such as factoring, completing the square, or the quadratic formula.

To get our equation into this form, we need to move all terms to one side of the equation, leaving zero on the other side. Let's subtract $8x$ and $8$ from both sides of the equation. This gives us:

$8x + 8 - 8x - 8 = x^2 + 10x - 8x - 8$

Simplifying both sides, we get:

$0 = x^2 + 2x - 8$

Now our equation is in the standard quadratic form: $x^2 + 2x - 8 = 0$ . Identifying $a$ , $b$ , and $c$ will be helpful for applying solution methods. In this case, $a = 1$ , $b = 2$ , and $c = -8$ . With the equation in this form, we are ready to proceed with solving for $x$ .

Step 3: Solve the Quadratic Equation

With the quadratic equation in the standard form $x^2 + 2x - 8 = 0$ , we have several options for solving it. We can try factoring, completing the square, or using the quadratic formula. In this case, factoring is the most straightforward method. We need to find two numbers that multiply to $-8$ and add up to $2$ . Those numbers are $4$ and $-2$ .

So, we can factor the quadratic equation as follows:

$(x + 4)(x - 2) = 0$

Now, according to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This means either $x + 4 = 0$ or $x - 2 = 0$ . Solving these two linear equations will give us the solutions for $x$ .

For $x + 4 = 0$ , we subtract $4$ from both sides to get $x = -4$ .

For $x - 2 = 0$ , we add $2$ to both sides to get $x = 2$ .

So, the solutions to the quadratic equation are $x = -4$ and $x = 2$ . These are the potential solutions to our original equation as well. However, we always need to check our solutions in the original equation to make sure they are valid.

Step 4: Check for Extraneous Solutions

After solving for $x$ , it's crucial to check our solutions in the original equation. This is especially important when dealing with rational equations because we may encounter extraneous solutions. Extraneous solutions are values that satisfy the transformed equation (in this case, the quadratic equation) but do not satisfy the original equation. They often arise due to the multiplication by a variable (like $x$ ) during the solving process.

Our original equation is $8 + \frac{8}{x} = x + 10$ , and our potential solutions are $x = -4$ and $x = 2$ . Let's check each one:

For $x = -4$ :

$8 + \frac{8}{-4} = -4 + 10$

$8 - 2 = 6$

$6 = 6$

This solution is valid.

For $x = 2$ :

$8 + \frac{8}{2} = 2 + 10$

$8 + 4 = 12$

$12 = 12$

This solution is also valid.

Since both $x = -4$ and $x = 2$ satisfy the original equation, they are our solutions. This step of verifying solutions is super important, guys, so never skip it!

Final Solution

After carefully working through each step, we have arrived at the final solution. We started with the equation $8 + \frac{8}{x} = x + 10$ , eliminated the fraction, rearranged the equation into quadratic form, solved for $x$ , and checked for extraneous solutions. We found that both $x = -4$ and $x = 2$ are valid solutions to the equation.

Therefore, the solutions to the equation $8 + \frac{8}{x} = x + 10$ are $x = -4$ and $x = 2$ .

Key Takeaways

Solving rational equations involves several key steps, and understanding each one is vital for success. Here are some key takeaways from our journey through this problem:

Eliminate the Fractions: Multiplying both sides of the equation by the common denominator is the first step to simplifying a rational equation.
Rearrange into Standard Form: For quadratic equations, rearranging into the form $ax^2 + bx + c = 0$ makes it easier to apply solution methods.
Solve the Quadratic: Factoring, completing the square, or the quadratic formula can be used to find the solutions.
Check for Extraneous Solutions: Always verify your solutions in the original equation to ensure they are valid.
Understanding the Domain: Be mindful of values that would make the denominator zero, as these are not valid solutions.

By mastering these steps, you'll be well-equipped to tackle a wide range of rational equations. Remember to practice consistently, and don't hesitate to review the steps when you encounter a challenging problem.

Practice Problems

Now that we've solved one equation together, it's time to put your skills to the test. Practice makes perfect, so try solving these similar equations on your own:

$\frac{12}{x} + 4 = x + 1$
$5 + \frac{6}{x} = x$
$\frac{10}{x} = x + 3$

Work through these problems using the steps we've discussed, and remember to check your solutions. If you get stuck, revisit the steps in this guide or seek help from a teacher or tutor. Keep practicing, and you'll become a pro at solving rational equations in no time! Guys, you've got this!