In this factorization, formula used: (x – 2y)² = x² – 4xy + y²

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} .