Solve the Equation: x/4y + y/x - 1 = 0 Using Multiplication Formulas

To solve this problem, we'll begin by simplifying the given expression:

The equation is $\frac{x}{4y} + \frac{y}{x} - 1 = 0$.

• Step 1: Eliminate the fraction by obtaining a common denominator for the left side of the equation. The common denominator is $4y \times x = 4xy$.
• Step 2: Rewrite each term with the common denominator:
  $\frac{x^2}{4xy} + \frac{4y^2}{4xy} - \frac{4y}{4y} = 0$. This becomes $\frac{x^2 + 4y^2 - 4xy}{4xy} = 0$.
• Step 3: Multiply through by $4xy$ to eliminate the denominator: $x^2 + 4y^2 - 4xy = 0$.
• Step 4: Recognize this as being equivalent to the expanded form of $(x - 2y)^2 = x^2 - 2xy + 4y^2 = 0$.
• Step 5: Confirm that $x^2 - 2xy + 4y^2 \equiv x^2 + 4y^2 - 4xy$, indicating $(x - 2y)^2 = 0$.

Thus, we conclude that the equation can be rewritten using the square of a difference formula:

Therefore, the short multiplication formula for the given equation is $(x - 2y)^2 = 0$.