Given the equation:

\[
\frac{1}{x - 4} + \frac{x}{x - 2} = \frac{2}{x^2 - 6x + 8}
\]

we need to solve for \( x \). Let's start by simplifying and solving the equation step by step.

### Step 1: Factorize the Denominator

First, notice that the denominator on the right side can be factorized:

\[
x^2 - 6x + 8 = (x - 4)(x - 2)
\]

So, the equation becomes:

\[
\frac{1}{x - 4} + \frac{x}{x - 2} = \frac{2}{(x - 4)(x - 2)}
\]

### Step 2: Combine the Left Side into a Single Fraction

To combine the fractions on the left side, find a common denominator:

\[
\frac{1}{x - 4} + \frac{x}{x - 2} = \frac{(x - 2) + x(x - 4)}{(x - 4)(x - 2)}
\]

Simplify the numerator:

\[
\frac{(x - 2) + x^2 - 4x}{(x - 4)(x - 2)} = \frac{x^2 - 3x - 2}{(x - 4)(x - 2)}
\]

### Step 3: Equate the Two Fractions

Now equate the two sides of the equation:

\[
\frac{x^2 - 3x - 2}{(x - 4)(x - 2)} = \frac{2}{(x - 4)(x - 2)}
\]

Since the denominators are the same, equate the numerators:

\[
x^2 - 3x - 2 = 2
\]

### Step 4: Solve the Quadratic Equation

Simplify the quadratic equation:

\[
x^2 - 3x - 4 = 0
\]

Factorize the equation:

\[
(x - 4)(x + 1) = 0
\]

So, the solutions are:

\[
x = 4 \quad \text{and} \quad x = -1
\]

### Step 5: Check for Restrictions

We must ensure that the values of \( x \) do not make the original denominators zero:

- \( x = 4 \) makes \( x - 4 = 0 \), which is not allowed.
- \( x = -1 \) does not violate any restrictions.

Thus, the only valid solution is:

\[
\boxed{x = -1}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you handle equations with multiple restrictions?
2. What happens when both potential solutions are restricted?
3. Can you demonstrate a similar problem with different factors?
4. What are the steps for factorizing a quadratic expression?
5. How do you verify solutions of rational equations?

**Tip:** Always check for restrictions on the variable when dealing with fractions; values that make the denominator zero are not valid!

Step-by-step solution to solving the rational equation 1/(x - 4) + x/(x - 2) = 2/(x^2 - 6x + 8). Learn how to factorize quadratic expressions and apply restrictions on denominators. This problem is ideal for students in Grades 9-11.

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