Simplify The Expression: 8 X 2 − 6 X − 16 + 9 X 2 − 3 X − 40 \frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40} X 2 − 6 X − 16 8 ​ + X 2 − 3 X − 40 9 ​

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is often required to solve various mathematical problems. In this article, we will focus on simplifying the given expression, which involves adding two fractions with quadratic denominators. The expression is $\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40}$. Our goal is to simplify this expression by finding a common denominator and combining the fractions.

Factorizing the Denominators

To simplify the given expression, we first need to factorize the denominators. The first denominator is $x^2 - 6x - 16$, and the second denominator is $x^2 - 3x - 40$. We can factorize these expressions as follows:

$x^2 - 6x - 16 = (x - 8)(x + 2)$

$x^2 - 3x - 40 = (x - 8)(x + 5)$

Finding a Common Denominator

Now that we have factorized the denominators, we can find a common denominator for the two fractions. The common denominator is the product of the two factorized expressions:

$(x - 8)(x + 2)(x - 8)(x + 5)$

However, we can simplify this expression further by canceling out the common factor $(x - 8)$:

$(x + 2)(x + 5)$

Simplifying the Expression

Now that we have found a common denominator, we can simplify the expression by combining the two fractions:

$\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40} = \frac{8(x + 5) + 9(x + 2)}{(x + 2)(x + 5)}$

Expanding the Numerator

To simplify the expression further, we need to expand the numerator:

$\frac{8(x + 5) + 9(x + 2)}{(x + 2)(x + 5)} = \frac{8x + 40 + 9x + 18}{(x + 2)(x + 5)}$

Combining Like Terms

Now that we have expanded the numerator, we can combine like terms:

$\frac{8x + 40 + 9x + 18}{(x + 2)(x + 5)} = \frac{17x + 58}{(x + 2)(x + 5)}$

Simplifying the Expression Further

We can simplify the expression further by dividing both the numerator and the denominator by their greatest common divisor, which is 1:

$\frac{17x + 58}{(x + 2)(x + 5)}$

Conclusion

In this article, we have simplified the given expression $\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40}$ by finding a common denominator and combining the fractions. The simplified expression is $\frac{17x + 58}{(x + 2)(x + 5)}$. This expression cannot be simplified further, and it is the final answer.

Final Answer

The final answer is $\boxed{\frac{17x + 58}{(x + 2)(x + 5)}}$.

Step-by-Step Solution

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

1. Factorize the denominators: $x^2 - 6x - 16 = (x - 8)(x + 2)$ and $x^2 - 3x - 40 = (x - 8)(x + 5)$.
2. Find a common denominator: $(x - 8)(x + 2)(x - 8)(x + 5)$.
3. Simplify the common denominator: $(x + 2)(x + 5)$.
4. Simplify the expression: $\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40} = \frac{8(x + 5) + 9(x + 2)}{(x + 2)(x + 5)}$.
5. Expand the numerator: $\frac{8(x + 5) + 9(x + 2)}{(x + 2)(x + 5)} = \frac{8x + 40 + 9x + 18}{(x + 2)(x + 5)}$.
6. Combine like terms: $\frac{8x + 40 + 9x + 18}{(x + 2)(x + 5)} = \frac{17x + 58}{(x + 2)(x + 5)}$.

Frequently Asked Questions

Here are some frequently asked questions related to the problem:

• Q: What is the simplified expression? A: The simplified expression is $\frac{17x + 58}{(x + 2)(x + 5)}$.
• Q: How do I simplify the expression? A: To simplify the expression, you need to find a common denominator and combine the fractions.
• Q: What is the final answer? A: The final answer is $\boxed{\frac{17x + 58}{(x + 2)(x + 5)}}$.

Related Problems

Here are some related problems that you may find useful:

• Simplify the expression: $\frac{3}{x^2 - 4x - 5} + \frac{2}{x^2 - 2x - 15}$
• Simplify the expression: $\frac{5}{x^2 - 2x - 6} + \frac{3}{x^2 - 5x - 12}$
• Simplify the expression: $\frac{2}{x^2 - 3x - 10} + \frac{4}{x^2 - 5x - 24}$
  

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Introduction

In our previous article, we simplified the expression $\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40}$ by finding a common denominator and combining the fractions. The simplified expression is $\frac{17x + 58}{(x + 2)(x + 5)}$. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the simplified expression?

A: The simplified expression is $\frac{17x + 58}{(x + 2)(x + 5)}$.

Q: How do I simplify the expression?

A: To simplify the expression, you need to find a common denominator and combine the fractions. You can factorize the denominators, find a common denominator, and then simplify the expression by combining the fractions.

Q: What is the final answer?

A: The final answer is $\boxed{\frac{17x + 58}{(x + 2)(x + 5)}}$.

Q: Can I simplify the expression further?

A: No, the expression $\frac{17x + 58}{(x + 2)(x + 5)}$ cannot be simplified further.

Q: What is the greatest common divisor of the numerator and the denominator?

A: The greatest common divisor of the numerator and the denominator is 1.

Q: How do I factorize the denominators?

A: To factorize the denominators, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, to factorize the denominator $x^2 - 6x - 16$, you need to find two numbers whose product is -16 and whose sum is -6. The numbers are -8 and 2, so the denominator can be factorized as $(x - 8)(x + 2)$.

Q: How do I find a common denominator?

A: To find a common denominator, you need to multiply the two factorized expressions. For example, to find a common denominator for the expressions $(x - 8)(x + 2)$ and $(x - 8)(x + 5)$, you need to multiply them together: $(x - 8)(x + 2)(x - 8)(x + 5)$.

Q: How do I simplify the expression by combining the fractions?

A: To simplify the expression by combining the fractions, you need to multiply the numerator and the denominator of each fraction by the other fraction's denominator. For example, to simplify the expression $\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40}$, you need to multiply the numerator and the denominator of each fraction by the other fraction's denominator: $\frac{8(x + 5) + 9(x + 2)}{(x + 2)(x + 5)}$.

Related Problems

Here are some related problems that you may find useful:

• Simplify the expression: $\frac{3}{x^2 - 4x - 5} + \frac{2}{x^2 - 2x - 15}$
• Simplify the expression: $\frac{5}{x^2 - 2x - 6} + \frac{3}{x^2 - 5x - 12}$
• Simplify the expression: $\frac{2}{x^2 - 3x - 10} + \frac{4}{x^2 - 5x - 24}$

Conclusion

In this article, we have answered some frequently asked questions related to the problem of simplifying the expression $\frac{8}{x^2 - 6x - 16} + \frac{9}{x^2 - 3x - 40}$. We have provided step-by-step solutions to the problem and have answered some common questions related to the problem. We hope that this article has been helpful in understanding the problem and its solution.

Final Answer

The final answer is $\boxed{\frac{17x + 58}{(x + 2)(x + 5)}}$.