Resolving the Equation y/x 2 - x/y and Its Implications on Algebraic Expressions

Introduction

In this article, we will explore a specific algebraic problem that involves solving an equation and then using the obtained result to simplify another complex expression. We will start by examining the given equation and solving it using algebraic techniques, and then we will apply the solution to find the value of the given expression. This problem will not only demonstrate the importance of algebraic manipulation but also provide insights into how finding the root of one problem can help solve more complex ones.

Solving the Equation y/x 2 - x/y

Given the equation:

$$frac{y}{x}2 - frac{x}{y}$$

We can begin by cross-multiplying to eliminate the fractions:

$$y^2 2xy - x^2$$

Next, we rearrange this into a standard quadratic equation:

$$y^2 - 2xy x^2 0$$

Using the quadratic formula to solve for y:

$$y frac{2x pm sqrt{(2x)^2 - 4(1)(x^2)}}{2(1)} frac{2x pm sqrt{4x^2 - 4x^2}}{2} frac{2x pm 0}{2} x$$

This shows that y x.

Evaluating the Expression with the Found Relationship

Now that we have established that y x, we can use this information to simplify the given expression:

$$frac{x^3xy^2}{x^3y^3}$$

Substituting y x into the expression:

$$frac{x^3 cdot x cdot x^2}{x^3 cdot x^3} frac{2x^3}{2x^3} 1$$

Therefore, the value of the given expression is:

$$1$$

Conclusion

In summary, by solving the equation y/x 2 - x/y and using the relationship found (i.e., y x), we have successfully simplified the expression to find its value. This process showcases the importance of algebraic manipulation and how one algebraic solution can be utilized to solve seemingly unrelated problems in mathematics.

Keywords

algebraic equation, quadratic formula, algebraic manipulation