Question

Express in simplest form:
x
2
−y
2

2y
2
−2xy
​
.

Answer 1

To simplify the expression \(\frac{x^2 - y^2}{2y^2 - 2xy}\), we can start by factoring both the numerator and the denominator.

**Step 1: Factor the numerator**

The numerator \(x^2 - y^2\) is a difference of squares, which can be factored as follows:

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

**Step 2: Factor the denominator**

The denominator \(2y^2 - 2xy\) has a common factor of \(2y\):

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

**Step 3: Rewrite the expression with the factored forms**

Substituting the factored forms back into the expression gives us:

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

**Step 4: Simplify the expression**

Notice that \(y - x\) can be rewritten as \(-(x - y)\):

\[
\frac{(x - y)(x + y)}{2y(y - x)} = \frac{(x - y)(x + y)}{2y(-1)(x - y)} = -\frac{(x + y)}{2y}
\]

Since \((x - y)\) cancels out from the numerator and denominator, we are left with:

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

**Final Answer:**

Thus, the simplest form of the expression is:

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