Add And Simplify. Leave The Denominator In Factored Form For The Final Answer. ${ \frac{4}{x^2-6x-40} + \frac{5}{x^2-12x+20} }$

Understanding Complex Fractions

Complex fractions are mathematical expressions that contain fractions within other fractions. They can be challenging to simplify, but with the right approach, you can break them down into more manageable parts. In this article, we will focus on simplifying a complex fraction involving two quadratic expressions in the denominator.

The Problem: Simplifying a Complex Fraction

The given complex fraction is:

$$\frac{4}{x^2-6x-40} + \frac{5}{x^2-12x+20}$$

Our goal is to simplify this expression and leave the denominator in factored form for the final answer.

Step 1: Factor the Denominators

To simplify the complex fraction, we need to factor the denominators of both fractions. Let's start by factoring the first denominator, $x^2-6x-40$. We can factor it as:

$$x^2-6x-40 = (x-8)(x+5)$$

Similarly, we can factor the second denominator, $x^2-12x+20$, as:

$$x^2-12x+20 = (x-10)(x-2)$$

Step 2: Rewrite the Complex Fraction with Factored Denominators

Now that we have factored the denominators, we can rewrite the complex fraction as:

$$\frac{4}{(x-8)(x+5)} + \frac{5}{(x-10)(x-2)}$$

Step 3: Find a Common Denominator

To add the two fractions, we need to find a common denominator. The least common multiple (LCM) of $(x-8)(x+5)$ and $(x-10)(x-2)$ is $(x-8)(x+5)(x-10)(x-2)$. We can rewrite the complex fraction with this common denominator as:

$$\frac{4(x-10)(x-2)}{(x-8)(x+5)(x-10)(x-2)} + \frac{5(x-8)(x+5)}{(x-8)(x+5)(x-10)(x-2)}$$

Step 4: Add the Numerators

Now that we have a common denominator, we can add the numerators:

$$\frac{4(x-10)(x-2) + 5(x-8)(x+5)}{(x-8)(x+5)(x-10)(x-2)}$$

Step 5: Simplify the Numerator

We can simplify the numerator by expanding and combining like terms:

$$\frac{4(x^2-12x+20) + 5(x^2-3x-40)}{(x-8)(x+5)(x-10)(x-2)}$$

$$\frac{4x^2-48x+80 + 5x^2-15x-200}{(x-8)(x+5)(x-10)(x-2)}$$

$$\frac{9x^2-63x-120}{(x-8)(x+5)(x-10)(x-2)}$$

Step 6: Factor the Numerator

We can factor the numerator as:

$$\frac{9(x^2-7x-13.33)}{(x-8)(x+5)(x-10)(x-2)}$$

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$$\frac{9(x-13.33)(x+1)}{(x-8)(x+5)(x-10)(x-2)}$$

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$$\frac{9(x-13)(x+1)}{(x-8)(x+5)(x-10)(x-2)}$$

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$$\frac{9(x-13)(x+1)}{(x-8)(x+5)(x-10)(x-2)}$$

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Understanding Complex Fractions

Complex fractions are mathematical expressions that contain fractions within other fractions. They can be challenging to simplify, but with the right approach, you can break them down into more manageable parts. In this article, we will focus on simplifying a complex fraction involving two quadratic expressions in the denominator.

The Problem: Simplifying a Complex Fraction

The given complex fraction is:

$$\frac{4}{x^2-6x-40} + \frac{5}{x^2-12x+20}$$

Our goal is to simplify this expression and leave the denominator in factored form for the final answer.

Step 1: Factor the Denominators

To simplify the complex fraction, we need to factor the denominators of both fractions. Let's start by factoring the first denominator, $x^2-6x-40$. We can factor it as:

$$x^2-6x-40 = (x-8)(x+5)$$

Similarly, we can factor the second denominator, $x^2-12x+20$, as:

$$x^2-12x+20 = (x-10)(x-2)$$

Step 2: Rewrite the Complex Fraction with Factored Denominators

Now that we have factored the denominators, we can rewrite the complex fraction as:

$$\frac{4}{(x-8)(x+5)} + \frac{5}{(x-10)(x-2)}$$

Step 3: Find a Common Denominator

To add the two fractions, we need to find a common denominator. The least common multiple (LCM) of $(x-8)(x+5)$ and $(x-10)(x-2)$ is $(x-8)(x+5)(x-10)(x-2)$. We can rewrite the complex fraction with this common denominator as:

$$\frac{4(x-10)(x-2)}{(x-8)(x+5)(x-10)(x-2)} + \frac{5(x-8)(x+5)}{(x-8)(x+5)(x-10)(x-2)}$$

Step 4: Add the Numerators

Now that we have a common denominator, we can add the numerators:

$$\frac{4(x-10)(x-2) + 5(x-8)(x+5)}{(x-8)(x+5)(x-10)(x-2)}$$

Step 5: Simplify the Numerator

We can simplify the numerator by expanding and combining like terms:

$$\frac{4(x^2-12x+20) + 5(x^2-3x-40)}{(x-8)(x+5)(x-10)(x-2)}$$

$$\frac{4x^2-48x+80 + 5x^2-15x-200}{(x-8)(x+5)(x-10)(x-2)}$$

$$\frac{9x^2-63x-120}{(x-8)(x+5)(x-10)(x-2)}$$

Step 6: Factor the Numerator

We can factor the numerator as:

$$\frac{9(x^2-7x-13.33)}{(x-8)(x+5)(x-10)(x-2)}$$

However, we can simplify this further by factoring the numerator as:

$$\frac{9(x-13)(x+1)}{(x-8)(x+5)(x-10)(x-2)}$$

Q&A

Q: What is a complex fraction?

A: A complex fraction is a mathematical expression that contains fractions within other fractions.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to factor the denominators, find a common denominator, add the numerators, and simplify the numerator.

Q: What is the least common multiple (LCM) of two expressions?

A: The least common multiple (LCM) of two expressions is the smallest expression that is a multiple of both expressions.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top part of a fraction, and the denominator is the bottom part of a fraction.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator and add the numerators.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

$$\frac{9(x-13)(x+1)}{(x-8)(x+5)(x-10)(x-2)}$$

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