(x^2-y^2)^2+4x^2y^2+x^2-2y^2=0

4 min read Jun 17, 2024

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.