Rationalize this equation (7/(x-8)=(x+1)/(x-8)+(x+1)/(x^2-8x))

To rationalize the given equation:

\[
\frac{7}{x-8} = \frac{x+1}{x-8} + \frac{x+1}{x^2 - 8x}
\]

we need to eliminate any fractions by finding a common denominator. 

### Step 1: Identify the denominators
The denominators in the equation are \( x-8 \) and \( x^2 - 8x \). Notice that \( x^2 - 8x \) can be factored as:

\[
x^2 - 8x = x(x - 8)
\]

### Step 2: Find the least common denominator (LCD)
The least common denominator of all terms is \( x(x - 8) \).

### Step 3: Multiply each term by the LCD
Multiply every term in the equation by \( x(x - 8) \) to eliminate the fractions:

\[
x(x - 8) \cdot \frac{7}{x-8} = x(x - 8) \cdot \frac{x+1}{x-8} + x(x - 8) \cdot \frac{x+1}{x(x-8)}
\]

### Step 4: Simplify each term
- For the first term:
\[
x(x - 8) \cdot \frac{7}{x-8} = 7x
\]
- For the second term:
\[
x(x - 8) \cdot \frac{x+1}{x-8} = x(x + 1)
\]
- For the third term:
\[
x(x - 8) \cdot \frac{x+1}{x(x-8)} = x + 1
\]

Now, the equation is:

\[
7x = x(x + 1) + (x + 1)
\]

### Step 5: Expand and simplify
Expand the terms on the right side:

\[
7x = x^2 + x + x + 1
\]

Combine like terms:

\[
7x = x^2 + 2x + 1
\]

### Step 6: Bring all terms to one side of the equation
Subtract \( 7x \) from both sides:

\[
0 = x^2 - 5x + 1
\]

This is the rationalized form of the original equation:

\[
x^2 - 5x + 1 = 0
\]

This quadratic equation can be solved further if required.

Would you like to go over the solution of this quadratic equation or do you have any questions?

Here are 5 related questions:
1. How do you solve the quadratic equation \(x^2 - 5x + 1 = 0\)?
2. What is the significance of rationalizing an equation?
3. Can you explain how to find the roots of the quadratic equation using the quadratic formula?
4. What are the steps involved in factoring a quadratic expression?
5. How do you simplify algebraic fractions with different denominators?

**Tip:** When rationalizing equations, always check if the denominators can be factored to find the least common denominator easily.

Learn how to rationalize the equation 7/(x-8) = (x+1)/(x-8) + (x+1)/(x^2 - 8x) step-by-step with a detailed solution. This problem involves equations, rationalization, and quadratic equations. Suitable for students in Grades 10-12.

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