simplify the rational expression (4x-8)/(x^2-4x+4)

The rational expression is:

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

### Step 1: Factor the numerator
The numerator \(4x - 8\) has a common factor of 4:

\[
4x - 8 = 4(x - 2)
\]

### Step 2: Factor the denominator
The denominator \(x^2 - 4x + 4\) is a perfect square trinomial:

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

### Step 3: Simplify the expression
Now, the rational expression becomes:

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

You can cancel out one factor of \(x - 2\) from the numerator and denominator (assuming \(x \neq 2\) to avoid division by zero):

\[
\frac{4}{x - 2}
\]

So, the simplified expression is:

\[
\frac{4}{x - 2} \quad \text{(for } x \neq 2\text{)}
\]

Would you like further details or have any questions?

Here are some related questions:
1. What is a perfect square trinomial, and how do you recognize it?
2. How do you factor more complex trinomials?
3. How do you handle rational expressions when there's no common factor?
4. What happens if \(x = 2\) in this expression?
5. How do you simplify complex rational expressions involving multiple terms?

**Tip**: Always check for common factors in both the numerator and denominator before simplifying a rational expression.

Learn how to simplify the rational expression (4x-8)/(x^2-4x+4) by factoring and using the perfect square trinomial formula. This solution walks through factoring, canceling common factors, and simplifying rational expressions. Suitable for students in Grades 8-10.

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