Question

Express in simplest form: 2y^2 - 2xy/x^2-y^2

Answer 1

To simplify the expression \(\frac{2y^2 - 2xy}{x^2 - y^2}\), we can follow these steps:

1. **Factor the numerator**:
The numerator \(2y^2 - 2xy\) can be factored by taking out the common factor \(2y\):

\[
2y^2 - 2xy = 2y(y - x)
\]

2. **Factor the denominator**:
The denominator \(x^2 - y^2\) is a difference of squares, which can be factored as:

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

3. **Rewriting the expression**:
Substituting the factored forms back into the expression gives:

\[
\frac{2y(y - x)}{(x - y)(x + y)}
\]

4. **Simplifying the expression**:
Notice that \((y - x) = -(x - y)\), so we can rewrite the numerator:

\[
2y(y - x) = -2y(x - y)
\]

Therefore, we have:

\[
\frac{-2y(x - y)}{(x - y)(x + y)}
\]

5. **Cancel the common factor**:
Assuming \(x \neq y\), we can cancel out \((x - y)\) from the numerator and denominator:

\[
= \frac{-2y}{x + y}
\]

Thus, the simplest form of the expression \( \frac{2y^2 - 2xy}{x^2 - y^2} \) is:

\[
\boxed{\frac{-2y}{x+y}}
\] (for \(x \neq y\)).