Problem:

Solve the rational equation:

$\frac{2}{x} = 1 + \frac{4x - 12}{x^2 - 2x}$

Step 1: Factor the denominator

First, let's factor the denominator on the right-hand side of the equation:

$x^2 - 2x = x(x - 2)$

This simplifies the equation to:

$\frac{2}{x} = 1 + \frac{4x - 12}{x(x - 2)}$

Step 2: Find the Least Common Denominator (LCD)

The denominators are $x$ and $x(x - 2)$. The Least Common Denominator (LCD) of the equation is:

$\text{LCD} = x(x - 2)$

Step 3: Multiply both sides by the LCD

Multiply both sides of the equation by the LCD $x(x - 2)$ to eliminate the denominators:

$x(x - 2) \cdot \frac{2}{x} = x(x - 2) \left( 1 + \frac{4x - 12}{x(x - 2)} \right)$

Simplifying both sides:

$2(x - 2) = x(x - 2) + 4x - 12$

Step 4: Simplify the equation

Expand both sides:

$2x - 4 = x^2 - 2x + 4x - 12$

Simplify the terms on both sides:

$2x - 4 = x^2 + 2x - 12$

Step 5: Rearrange the equation to form a quadratic equation

Move all terms to one side of the equation:

$0 = x^2 + 2x - 12 - 2x + 4$

Simplify:

$0 = x^2 - 8$

Step 6: Solve the quadratic equation

Solve for $x$:

$x^2 = 8$

$x = \pm \sqrt{8}$

Simplify:

$x = \pm 2\sqrt{2}$

Step 7: Check for extraneous solutions

We must check the solutions in the original equation to ensure they are valid. The denominator cannot be zero, so check when $x = 0$ or $x = 2$, as these would make the denominator zero.

The solutions $x = \pm 2\sqrt{2}$ are valid, as they do not make the denominator zero.

Final Answer:

$x = 2\sqrt{2}, \quad x = -2\sqrt{2}$

Checking:

Substitute $x = 2\sqrt{2}$ and $x = -2\sqrt{2}$ into the original equation and verify that both sides of the equation hold true. This process confirms the solutions are correct.

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Would you like a more detailed explanation of the checking step or any part of this process?

Here are five related problems for practice:

1. Solve $\frac{3}{x} = 1 + \frac{5x - 10}{x^2 - x}$.
2. Solve $\frac{4}{x} = 2 + \frac{6x - 18}{x^2 - 3x}$.
3. Solve $\frac{1}{x} = 2 + \frac{3x - 9}{x(x - 3)}$.
4. Solve $\frac{2}{x + 1} = 3 + \frac{7x - 14}{(x + 1)^2}$.
5. Solve $\frac{5}{x} = 4 + \frac{6x - 24}{x(x - 4)}$.

Tip: Always check your solutions by substituting them back into the original equation to avoid extraneous solutions!