In this factorization, formula used: (x – 2y)^{2}= x^{2}– 4xy + y^{2}

**Explanation**

Let’s solve the given equation first:

\rightarrow\ \frac{x}{y}\ +\ \frac{y}{x} = 4

To simplify this equation:

\rightarrow\ \frac{x^2\ +\ y^2}{xy} = 4

\rightarrow\ x^2 + y^2 = 4xy _{[cross multiply]}

Now, let’s rearrange the terms:

\rightarrow\ x^2 – 4xy + y^2 = 0

Factor the quadratic expression:

\rightarrow (x – 2y)^2 =0

\rightarrow\ \left(x\ -\ 2y\right)\ =\ \sqrt0

\rightarrow (x – 2y) = 0

\rightarrow\ x\ =\ 2y

So, x = 2y is the solution to the given equation.

Now, let’s find \ \frac{x}{y}\ – \ \frac{y}{x}\ using the value x = 2y.

\rightarrow\ \frac{x}{y}\ – \frac{y}{x}\

\rightarrow\ \frac{2y}{y}\ – \frac{y}{2y}\

Simplify each term:

\rightarrow\ 2 – \frac{1}{2}

Combine the terms:

\rightarrow\ \frac{4}{2}\ – \frac{1}{2}\

\rightarrow\ \frac{4\ -\ 1}{2}

\rightarrow\ \frac{3}{2}

So, \frac{x}{y}\ – \frac{y}{x}\ = \frac{3}{2} when x = 2y. **Ans:** \frac{x}{y}\ -\ \frac{y}{x}\ =\ \frac{3}{2} .