Exploring the Equation: (x^2  y^2)^2 + 4x^2y^2 + x^2  2y^2 = 0
This equation, at first glance, appears complex. However, with a few algebraic manipulations and a keen eye for patterns, we can unveil its secrets and understand its nature.
Simplifying the Expression

Expanding the Square: Let's begin by expanding the first term: (x^2  y^2)^2 = x^4  2x^2y^2 + y^4

Combining Terms: Now, substitute this expansion back into the original equation and group similar terms: x^4  2x^2y^2 + y^4 + 4x^2y^2 + x^2  2y^2 = 0 x^4 + 2x^2y^2 + y^4 + x^2  2y^2 = 0

Recognizing Patterns: Notice that the first three terms (x^4 + 2x^2y^2 + y^4) form a perfect square: (x^2 + y^2)^2. The last two terms (x^2  2y^2) can be factored as (x^2  2y^2)

Simplified Form: Now we have: (x^2 + y^2)^2 + (x^2  2y^2) = 0
Analyzing the Equation
The simplified form reveals that the equation represents a relationship between x and y. It states that the sum of two squares must equal zero. This has a crucial implication:
Both squares must be equal to zero.
This leads to two equations:
 (x^2 + y^2)^2 = 0 => x^2 + y^2 = 0
 (x^2  2y^2) = 0
Finding Solutions

Equation 1: x^2 + y^2 = 0. The sum of two squares can only be zero if both squares are zero. Therefore, x = 0 and y = 0.

Equation 2: x^2  2y^2 = 0. This equation tells us x^2 = 2y^2. Since we already know x = 0 and y = 0, this equation is satisfied.
Conclusion
The equation (x^2  y^2)^2 + 4x^2y^2 + x^2  2y^2 = 0 has only one solution: x = 0 and y = 0. This solution is a point on the coordinate plane, representing the origin (0, 0).
In essence, the equation describes a very specific relationship between x and y, where both variables must be zero to satisfy the equation.