Solving Complex Equations Using Basic Algebraic Identities: x2 y2 73 and xy 24

In this article, we explore a method to solve a complex equation problem using fundamental algebraic identities. The problem given is to find the value of ( x - y^2 ), given the equations ( x^2y^2 73 ) and ( xy 24 ). We will walk through the solution step-by-step, demonstrating how to effectively utilize algebraic identities to solve for the desired variable.

Problem Statement and Given Information

Given Equations: ( x^2y^2 73 ) ( xy 24 ) Objective: Find the value of ( x - y^2 ).

Step-by-Step Solution

Our goal is to manipulate the given equations to find an expression for ( x - y^2 ). We'll use algebraic identities, particularly the identity for the difference of squares, to solve the problem.

Step 1: Formulate the Identity for ( x - y^2 )

Recall the algebraic identity: [x - y^2 x^2y^2 - 2xy] Substituting the given values, we can use this identity to find ( x - y^2 ). Let's proceed step-by-step.

Step 2: Substitute the Known Values

First, substitute the given values into the identity: [x - y^2 x^2y^2 - 2xy] We know from the given equations that: [x^2y^2 73 quad text{and} quad xy 24] Thus, substituting these values into the identity: [x - y^2 73 - 2 cdot 24]

Step 3: Simplify the Expression

Now, perform the multiplication and subtraction: [x - y^2 73 - 48 25] Therefore, the value of ( x - y^2 ) is: [boxed{25}]

Conclusion

The key to solving this problem was recognizing and applying the algebraic identity correctly. By using the identity ( x - y^2 x^2y^2 - 2xy ), we were able to directly substitute the given values to find the required expression.

Additional Insights

The solutions for ( xy ) can be further explored to understand the underlying quadratic equations. However, this tailored approach demonstrates the effectiveness of basic algebraic identities in solving more complex problems.

Explore Further

For those interested in deeper exploration, consider the following topics:

Understanding and applying other algebraic identities such as the difference of squares, sum of squares, and others. Solving more complex systems of equations using substitution and elimination methods. Exploring the relationship between roots and coefficients in quadratic equations.

By mastering these foundational algebraic concepts, you can tackle a variety of mathematical challenges with confidence.