Let's solve the expression step by step:

Given expression:
\[
\frac{w^2}{w^2 - 4} - \frac{1}{w - 2}
\]

### Step 1: Factor the denominator of the first fraction
The denominator \( w^2 - 4 \) is a difference of squares, which can be factored as:
\[
w^2 - 4 = (w - 2)(w + 2)
\]
So, the expression becomes:
\[
\frac{w^2}{(w - 2)(w + 2)} - \frac{1}{w - 2}
\]

### Step 2: Rewrite the second fraction
To combine the fractions, we need a common denominator. The common denominator will be \((w - 2)(w + 2)\). So, rewrite the second fraction with the common denominator:
\[
\frac{1}{w - 2} = \frac{(w + 2)}{(w - 2)(w + 2)}
\]
The expression now is:
\[
\frac{w^2}{(w - 2)(w + 2)} - \frac{(w + 2)}{(w - 2)(w + 2)}
\]

### Step 3: Combine the fractions
Since both fractions now have the same denominator, combine them:
\[
\frac{w^2 - (w + 2)}{(w - 2)(w + 2)}
\]

### Step 4: Simplify the numerator
Simplify the expression in the numerator:
\[
w^2 - (w + 2) = w^2 - w - 2
\]

So the final expression is:
\[
\frac{w^2 - w - 2}{(w - 2)(w + 2)}
\]

### Step 5: Factor the numerator (if possible)
The numerator \( w^2 - w - 2 \) can be factored as:
\[
w^2 - w - 2 = (w - 2)(w + 1)
\]

Thus, the expression simplifies to:
\[
\frac{(w - 2)(w + 1)}{(w - 2)(w + 2)}
\]

### Step 6: Cancel common factors
Since \( w - 2 \) is a common factor in the numerator and denominator, it can be canceled out (assuming \( w \neq 2 \) to avoid division by zero):
\[
\frac{w + 1}{w + 2}
\]

### Final Answer:
\[
\boxed{\frac{w + 1}{w + 2}}
\]

Would you like more details or have any questions? Here are some related topics:

1. How to factor polynomials.
2. Understanding the difference of squares.
3. Rules for simplifying rational expressions.
4. Dealing with undefined expressions in algebra.
5. How to find common denominators in algebraic fractions.

**Tip:** Always check for common factors in the numerator and denominator before simplifying. This helps avoid unnecessary complications.

Learn how to simplify the rational expression (w^2)/(w^2 - 4) - (1)/(w - 2) using algebraic techniques. This step-by-step solution covers factoring, finding common denominators, and simplifying the final expression. Suitable for Grades 9-10.

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