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

1 answer

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 \(x \neq y\)).