Simplest form of

Question

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:

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