Simplify The Expression:$\[ \frac{x}{x+4}-\frac{x+36}{x^2-16} \\]

Introduction

When dealing with algebraic expressions, simplifying them is an essential step in solving equations and inequalities. In this article, we will focus on simplifying a given expression that involves combining fractions. The expression we will be working with is $\frac{x}{x+4}-\frac{x+36}{x^2-16}$. Our goal is to simplify this expression and provide a clear understanding of the steps involved.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at it. The expression consists of two fractions: $\frac{x}{x+4}$ and $\frac{x+36}{x^2-16}$. The first fraction has a numerator of $x$ and a denominator of $x+4$. The second fraction has a numerator of $x+36$ and a denominator of $x^2-16$.

Simplifying the Denominator

To simplify the expression, we need to start by simplifying the denominator of the second fraction. The denominator is $x^2-16$, which can be factored as $(x+4)(x-4)$. This gives us $\frac{x+36}{(x+4)(x-4)}$.

Combining the Fractions

Now that we have simplified the denominator, we can combine the two fractions. To do this, we need to find a common denominator, which is the product of the two denominators: $(x+4)(x-4)$. We can rewrite the first fraction with this common denominator: $\frac{x(x-4)}{(x+4)(x-4)}$.

Simplifying the Expression

Now that we have combined the fractions, we can simplify the expression. We can start by simplifying the numerator: $x(x-4) - (x+36)$. This gives us $x^2 - 4x - x - 36$, which simplifies to $x^2 - 5x - 36$.

Factoring the Numerator

The numerator $x^2 - 5x - 36$ can be factored as $(x-9)(x+4)$. This gives us $\frac{(x-9)(x+4)}{(x+4)(x-4)}$.

Canceling Common Factors

Now that we have factored the numerator, we can cancel out common factors between the numerator and the denominator. The common factor is $(x+4)$, which can be canceled out. This gives us $\frac{x-9}{x-4}$.

Conclusion

In this article, we have simplified the expression $\frac{x}{x+4}-\frac{x+36}{x^2-16}$ by combining the fractions and canceling out common factors. The simplified expression is $\frac{x-9}{x-4}$. This expression can be further simplified by factoring the numerator and canceling out common factors.

Final Answer

The final answer is $\boxed{\frac{x-9}{x-4}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the denominator of the second fraction: $\frac{x+36}{x^2-16} = \frac{x+36}{(x+4)(x-4)}$
2. Combine the fractions: $\frac{x}{x+4} - \frac{x+36}{(x+4)(x-4)} = \frac{x(x-4)}{(x+4)(x-4)} - \frac{x+36}{(x+4)(x-4)}$
3. Simplify the numerator: $x(x-4) - (x+36) = x^2 - 4x - x - 36 = x^2 - 5x - 36$
4. Factor the numerator: $x^2 - 5x - 36 = (x-9)(x+4)$
5. Cancel out common factors: $\frac{(x-9)(x+4)}{(x+4)(x-4)} = \frac{x-9}{x-4}$

Tips and Tricks

• When simplifying expressions, it's essential to start by simplifying the denominator.
• Combining fractions requires finding a common denominator.
• Canceling out common factors can help simplify the expression.
• Factoring the numerator can help identify common factors to cancel out.

Real-World Applications

Simplifying expressions is a crucial skill in mathematics, and it has numerous real-world applications. In physics, for example, simplifying expressions can help solve equations that describe the motion of objects. In engineering, simplifying expressions can help design and optimize systems. In finance, simplifying expressions can help calculate interest rates and investment returns.

Conclusion

In conclusion, simplifying the expression $\frac{x}{x+4}-\frac{x+36}{x^2-16}$ requires a step-by-step approach. By simplifying the denominator, combining the fractions, and canceling out common factors, we can arrive at the simplified expression $\frac{x-9}{x-4}$. This expression can be further simplified by factoring the numerator and canceling out common factors.

Introduction

In our previous article, we simplified the expression $\frac{x}{x+4}-\frac{x+36}{x^2-16}$ by combining the fractions and canceling out common factors. In this article, we will provide a Q&A guide to help you understand the steps involved in simplifying the expression.

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to simplify the denominator of the second fraction. This involves factoring the denominator to find its prime factors.

Q: How do I simplify the denominator of the second fraction?

A: To simplify the denominator of the second fraction, you need to factor the expression $x^2-16$. This can be done by finding two numbers whose product is $-16$ and whose sum is $0$. The two numbers are $4$ and $-4$, so the denominator can be factored as $(x+4)(x-4)$.

Q: What is the next step in simplifying the expression?

A: The next step in simplifying the expression is to combine the two fractions. This involves finding a common denominator, which is the product of the two denominators: $(x+4)(x-4)$.

Q: How do I combine the fractions?

A: To combine the fractions, you need to rewrite the first fraction with the common denominator. This involves multiplying the numerator and denominator of the first fraction by $(x-4)$.

Q: What is the simplified expression after combining the fractions?

A: After combining the fractions, the simplified expression is $\frac{x(x-4)}{(x+4)(x-4)} - \frac{x+36}{(x+4)(x-4)}$.

Q: How do I simplify the numerator of the expression?

A: To simplify the numerator of the expression, you need to combine the two terms: $x(x-4)$ and $-(x+36)$. This involves combining like terms and simplifying the expression.

Q: What is the simplified numerator of the expression?

A: The simplified numerator of the expression is $x^2 - 5x - 36$.

Q: How do I factor the numerator of the expression?

A: To factor the numerator of the expression, you need to find two numbers whose product is $-36$ and whose sum is $-5$. The two numbers are $-9$ and $-4$, so the numerator can be factored as $(x-9)(x+4)$.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{(x-9)(x+4)}{(x+4)(x-4)}$.

Q: How do I cancel out common factors in the expression?

A: To cancel out common factors in the expression, you need to identify the common factors between the numerator and the denominator. In this case, the common factor is $(x+4)$.

Q: What is the final simplified expression after canceling out common factors?

A: The final simplified expression after canceling out common factors is $\frac{x-9}{x-4}$.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has numerous real-world applications in physics, engineering, and finance. In physics, simplifying expressions can help solve equations that describe the motion of objects. In engineering, simplifying expressions can help design and optimize systems. In finance, simplifying expressions can help calculate interest rates and investment returns.

Q: What are some tips and tricks for simplifying expressions?

A: Some tips and tricks for simplifying expressions include:

• Simplifying the denominator first
• Combining fractions requires finding a common denominator
• Canceling out common factors can help simplify the expression
• Factoring the numerator can help identify common factors to cancel out

Conclusion

In conclusion, simplifying the expression $\frac{x}{x+4}-\frac{x+36}{x^2-16}$ requires a step-by-step approach. By simplifying the denominator, combining the fractions, and canceling out common factors, we can arrive at the simplified expression $\frac{x-9}{x-4}$. This expression can be further simplified by factoring the numerator and canceling out common factors.