Question

Simplest form of
y(2y-2x) over x^2-y^2

Answer 1

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

1. **Factor the numerator and the denominator.**

We start with the numerator:
\[
y(2y - 2x) = 2y(y - x)
\]

Next, we factor the denominator:
\[
x^2 - y^2 = (x - y)(x + y) \quad \text{(This is a difference of squares)}
\]

2. **Substitute the factored forms back into the expression:**
\[
\frac{2y(y - x)}{(x - y)(x + y)}
\]

Now, notice that \(y - x\) can be rewritten as \(-(x - y)\):
\[
y - x = -(x - y)
\]

So we can substitute this into our expression:
\[
\frac{2y(-(x - y))}{(x - y)(x + y)} = \frac{-2y(x - y)}{(x - y)(x + y)}
\]

3. **Cancel the common factor \((x - y)\) (assuming \(x \neq y\)) from the numerator and denominator:**
\[
\frac{-2y}{x + y}
\]

4. **Final simplified form:**
\[
-\frac{2y}{x + y}
\]

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

\[
-\frac{2y}{x + y}, \quad \text{for } x \neq y.
\]