To solve the problem of adding the fractions \(\frac{w}{w - 2}\) and \(\frac{w + 2}{w}\), let's follow these steps:
1. Identify the common denominator:
The denominators are \(w - 2\) and \(w\). The common denominator is their product: \(w(w - 2)\).
2. Rewrite each fraction with the common denominator:
[tex]\[
\frac{w}{w - 2} = \frac{w \cdot w}{(w - 2) \cdot w} = \frac{w^2}{w(w - 2)}
\][/tex]
[tex]\[
\frac{w + 2}{w} = \frac{(w + 2) \cdot (w - 2)}{w \cdot (w - 2)} = \frac{(w + 2)(w - 2)}{w(w - 2)}
\][/tex]
3. Expand the terms in the second fraction:
[tex]\[
(w + 2)(w - 2) = w^2 - 4
\][/tex]
So,
[tex]\[
\frac{(w + 2)(w - 2)}{w(w - 2)} = \frac{w^2 - 4}{w(w - 2)}
\][/tex]
4. Combine the fractions:
[tex]\[
\frac{w^2}{w(w - 2)} + \frac{w^2 - 4}{w(w - 2)} = \frac{w^2 + (w^2 - 4)}{w(w - 2)}
\][/tex]
5. Simplify the numerator:
[tex]\[
w^2 + (w^2 - 4) = w^2 + w^2 - 4 = 2w^2 - 4
\][/tex]
6. Write the combined fraction:
[tex]\[
\frac{2w^2 - 4}{w(w - 2)}
\][/tex]
7. Factor the numerator:
Notice that \(2w^2 - 4\) can be factored:
[tex]\[
2w^2 - 4 = 2(w^2 - 2)
\][/tex]
So,
[tex]\[
\frac{2(w^2 - 2)}{w(w - 2)}
\][/tex]
So the simplified result of the given expression is:
[tex]\[
\frac{2(w^2 - 2)}{w(w - 2)}
\][/tex]
This is the simplified form of the expression [tex]\(\frac{w}{w-2} + \frac{w+2}{w}\)[/tex].
