Solve For All Values Of $x$. $x-\frac{2}{x-8}=7$ Answer: $ X = X= X = [/tex] $\square$

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Introduction

In mathematics, solving equations is a fundamental concept that helps us understand the relationships between variables. However, when we encounter complex equations, it can be challenging to find the solution. In this article, we will focus on solving a specific equation: $x-\frac{2}{x-8}=7$. We will break down the solution into manageable steps, making it easier to understand and follow.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The equation is a rational equation, which means it contains fractions with variables in the numerator and denominator. The equation is also a linear equation, as it can be written in the form $ax+b=c$, where $a$, $b$, and $c$ are constants.

Step 1: Multiply Both Sides by the Denominator

To eliminate the fraction, we need to multiply both sides of the equation by the denominator, which is $x-8$. This will help us get rid of the fraction and make the equation easier to work with.

(x−2x−8)(x−8)=7(x−8)\left(x-\frac{2}{x-8}\right)(x-8)=7(x-8)

Step 2: Distribute and Simplify

Now that we have multiplied both sides by the denominator, we can distribute and simplify the equation.

x(x−8)−2=7(x−8)x(x-8)-2=7(x-8)

x2−8x−2=7x−56x^2-8x-2=7x-56

Step 3: Move All Terms to One Side

To solve for $x$, we need to move all the terms to one side of the equation. Let's move all the terms to the left-hand side.

x2−8x−2−7x+56=0x^2-8x-2-7x+56=0

x2−15x+54=0x^2-15x+54=0

Step 4: Factor the Quadratic Equation

Now that we have a quadratic equation, we can try to factor it. If we can factor the equation, we can find the values of $x$ that satisfy the equation.

(x−9)(x−6)=0(x-9)(x-6)=0

Step 5: Solve for $x$

Now that we have factored the quadratic equation, we can solve for $x$. We can set each factor equal to zero and solve for $x$.

x−9=0⇒x=9x-9=0 \Rightarrow x=9

x−6=0⇒x=6x-6=0 \Rightarrow x=6

Conclusion

In this article, we have solved a complex equation using a step-by-step approach. We started by multiplying both sides of the equation by the denominator, then distributed and simplified the equation. We moved all the terms to one side of the equation and factored the quadratic equation. Finally, we solved for $x$ by setting each factor equal to zero. The solutions to the equation are $x=9$ and $x=6$.

Final Answer

Q&A: Solving a Complex Equation

In the previous article, we solved a complex equation using a step-by-step approach. However, we understand that some readers may still have questions about the solution. In this article, we will address some of the most frequently asked questions about solving a complex equation.

Q: What is a complex equation?

A complex equation is an equation that contains fractions with variables in the numerator and denominator. It can also be a linear equation, as it can be written in the form $ax+b=c$, where $a$, $b$, and $c$ are constants.

Q: Why do we need to multiply both sides by the denominator?

We need to multiply both sides by the denominator to eliminate the fraction. This will help us get rid of the fraction and make the equation easier to work with.

Q: How do we distribute and simplify the equation?

To distribute and simplify the equation, we need to multiply each term in the equation by the denominator. This will help us get rid of the fraction and make the equation easier to work with.

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means it can be written in the form $ax^2+bx+c=0$, where $a$, $b$, and $c$ are constants.

Q: How do we factor a quadratic equation?

To factor a quadratic equation, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. We can then write the quadratic equation as a product of two binomials.

Q: What are the solutions to the equation?

The solutions to the equation are the values of $x$ that satisfy the equation. In this case, the solutions are $x=9$ and $x=6$.

Q: Why do we need to check our solutions?

We need to check our solutions to make sure they are correct. We can do this by plugging the solutions back into the original equation and checking if they satisfy the equation.

Q: What are some common mistakes to avoid when solving a complex equation?

Some common mistakes to avoid when solving a complex equation include:

  • Not multiplying both sides by the denominator
  • Not distributing and simplifying the equation
  • Not factoring the quadratic equation correctly
  • Not checking the solutions

Conclusion

In this article, we have addressed some of the most frequently asked questions about solving a complex equation. We have covered topics such as what a complex equation is, why we need to multiply both sides by the denominator, and how to factor a quadratic equation. We have also discussed some common mistakes to avoid when solving a complex equation.

Final Answer

The final answer is: 9,6\boxed{9,6}