## Solving the Equation: (x^2 - 2x)(2x - 2) - 9(2x - 2) / (x^2 - 2x) = 0

This equation presents a challenge due to its complex structure and the presence of fractions. Let's break down the steps to find the solution:

### 1. Factoring and Simplifying

**Factor out (2x - 2):**Notice that (2x - 2) appears in both terms of the numerator. Factoring it out simplifies the expression: (2x - 2) [(x^2 - 2x) - 9 / (x^2 - 2x)] = 0**Simplify the bracketed term:**To combine the terms within the brackets, we need a common denominator. The common denominator is (x^2 - 2x): (2x - 2) [(x^2 - 2x)^2 - 9] / (x^2 - 2x) = 0**Further simplification:**Now, we can simplify the numerator: (2x - 2) [(x^2 - 2x + 3)(x^2 - 2x - 3)] / (x^2 - 2x) = 0

### 2. Identifying Potential Solutions

**Zero Product Property:**The product of two factors is zero if and only if at least one of the factors is zero. We can apply this property to our equation:- 2x - 2 = 0
- x^2 - 2x + 3 = 0
- x^2 - 2x - 3 = 0
- x^2 - 2x = 0

### 3. Solving for x

**Solve for x in 2x - 2 = 0:**- 2x = 2
**x = 1**

**Solve for x in x^2 - 2x + 3 = 0:**- This quadratic equation does not factor easily. We can use the quadratic formula:
- x = [-b ± √(b^2 - 4ac)] / 2a
- In this case, a = 1, b = -2, and c = 3.
- x = [2 ± √((-2)^2 - 4 * 1 * 3)] / 2 * 1
- x = [2 ± √(-8)] / 2
**x = 1 ± i√2**(where 'i' is the imaginary unit)

- This quadratic equation does not factor easily. We can use the quadratic formula:
**Solve for x in x^2 - 2x - 3 = 0:**- This quadratic equation factors:
- (x - 3)(x + 1) = 0
**x = 3**or**x = -1**

- This quadratic equation factors:
**Solve for x in x^2 - 2x = 0:**- Factor out x:
- x(x - 2) = 0
**x = 0**or**x = 2**

- Factor out x:

### 4. Checking for Extraneous Solutions

It's important to check if any of the potential solutions make the denominator (x^2 - 2x) equal to zero, as this would make the original equation undefined.

**x = 0**and**x = 2**make the denominator zero. Therefore, these solutions are extraneous and must be discarded.

### 5. Final Solutions

The solutions to the equation (x^2 - 2x)(2x - 2) - 9(2x - 2) / (x^2 - 2x) = 0 are:

**x = 1****x = 1 + i√2****x = 1 - i√2****x = 3****x = -1**