Solve The Equation For X X X : 1 X − 5 = 1 X + 3 + X − 1 X 2 − 2 X − 15 \frac{1}{x-5}=\frac{1}{x+3}+\frac{x-1}{x^2-2x-15} X − 5 1 ​ = X + 3 1 ​ + X 2 − 2 X − 15 X − 1 ​ A. X = 1 X = 1 X = 1 B. X = 5 , − 3 X = 5, -3 X = 5 , − 3 C. X = ± 9 X = \pm 9 X = ± 9 D. X = 9 X = 9 X = 9

Introduction

In this article, we will delve into solving a complex equation involving fractions and quadratic expressions. The given equation is $\frac{1}{x-5}=\frac{1}{x+3}+\frac{x-1}{x^2-2x-15}$. Our goal is to find the value of $x$ that satisfies this equation. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Simplify the Right-Hand Side of the Equation

To begin solving the equation, we need to simplify the right-hand side by finding a common denominator for the two fractions. The common denominator is $(x+3)(x-5)$.

\frac{1}{x+3}+\frac{x-1}{x^2-2x-15} = \frac{(x+3)+(x-1)(x-5)}{(x+3)(x-5)}

Step 2: Expand and Simplify the Numerator

Next, we expand and simplify the numerator of the fraction.

(x+3)+(x-1)(x-5) = x+3+x^2-5x-x+5 = x^2-3x+8

Step 3: Rewrite the Equation with the Simplified Right-Hand Side

Now that we have simplified the right-hand side, we can rewrite the equation as follows:

\frac{1}{x-5} = \frac{x^2-3x+8}{(x+3)(x-5)}

Step 4: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is $(x+3)(x-5)$.

(x+3)(x-5) \cdot \frac{1}{x-5} = (x+3)(x-5) \cdot \frac{x^2-3x+8}{(x+3)(x-5)}

Step 5: Simplify the Equation

After multiplying both sides by the LCM, we can simplify the equation by canceling out the common factors.

x+3 = x^2-3x+8

Step 6: Rearrange the Equation

To make it easier to solve, we can rearrange the equation by moving all the terms to one side.

x^2-4x+5 = 0

Step 7: Solve the Quadratic Equation

Now we have a quadratic equation in the form $ax^2+bx+c=0$. We can solve it using the quadratic formula or factoring.

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

In this case, we can factor the quadratic expression.

x^2-4x+5 = (x-5)(x-1) = 0

Step 8: Find the Values of x

Finally, we can find the values of $x$ that satisfy the equation by setting each factor equal to zero.

x-5 = 0 \Rightarrow x = 5
x-1 = 0 \Rightarrow x = 1

Conclusion

In conclusion, we have successfully solved the equation $\frac{1}{x-5}=\frac{1}{x+3}+\frac{x-1}{x^2-2x-15}$ and found the values of $x$ that satisfy the equation. The correct answer is $x = 5, -3$.

Answer Key

A. $x = 1$ B. $x = 5, -3$ C. $x = \pm 9$ D. $x = 9$

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Introduction

In our previous article, we solved the equation $\frac{1}{x-5}=\frac{1}{x+3}+\frac{x-1}{x^2-2x-15}$ and found the values of $x$ that satisfy the equation. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify the right-hand side by finding a common denominator for the two fractions.

Q: How do I find the common denominator?

A: To find the common denominator, we need to multiply the denominators of the two fractions together. In this case, the common denominator is $(x+3)(x-5)$.

Q: What is the next step in solving the equation?

A: After finding the common denominator, we need to expand and simplify the numerator of the fraction.

Q: How do I expand and simplify the numerator?

A: To expand and simplify the numerator, we need to multiply the terms inside the parentheses and combine like terms. In this case, the numerator simplifies to $x^2-3x+8$.

Q: What is the next step in solving the equation?

A: After simplifying the numerator, we need to rewrite the equation with the simplified right-hand side.

Q: How do I eliminate the fractions?

A: To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is $(x+3)(x-5)$.

Q: What is the next step in solving the equation?

A: After eliminating the fractions, we need to simplify the equation by canceling out the common factors.

Q: How do I simplify the equation?

A: To simplify the equation, we need to move all the terms to one side and combine like terms. In this case, the equation simplifies to $x^2-4x+5 = 0$.

Q: What is the next step in solving the equation?

A: After simplifying the equation, we need to solve the quadratic equation.

Q: How do I solve the quadratic equation?

A: To solve the quadratic equation, we can use the quadratic formula or factoring. In this case, we can factor the quadratic expression.

Q: What is the final step in solving the equation?

A: After factoring the quadratic expression, we need to find the values of $x$ that satisfy the equation by setting each factor equal to zero.

Q: What are the values of x that satisfy the equation?

A: The values of $x$ that satisfy the equation are $x = 5, -3$.

Conclusion

In conclusion, we have provided a Q&A guide to help you understand the solution to the equation $\frac{1}{x-5}=\frac{1}{x+3}+\frac{x-1}{x^2-2x-15}$. We hope this guide has been helpful in answering any questions you may have had.

Frequently Asked Questions

• Q: What is the first step in solving the equation? A: The first step in solving the equation is to simplify the right-hand side by finding a common denominator for the two fractions.
• Q: How do I find the common denominator? A: To find the common denominator, we need to multiply the denominators of the two fractions together.
• Q: What is the next step in solving the equation? A: After finding the common denominator, we need to expand and simplify the numerator of the fraction.

Additional Resources

• For more information on solving quadratic equations, please see our article on "Solving Quadratic Equations: A Step-by-Step Guide".
• For more information on factoring quadratic expressions, please see our article on "Factoring Quadratic Expressions: A Step-by-Step Guide".

Answer Key

A. $x = 1$ B. $x = 5, -3$ C. $x = \pm 9$ D. $x = 9$

The correct answer is B. $x = 5, -3$.