CPhill: I think that you graphed 2x^2 + 4xy + 2y^2 = 0, not 2x^2 + 4xy + 4y^2 = 0
Looking at 2x² + 4xy + 4y² = 0
Divide by 2: x² + 2xy + 2y² = 0
Solving for y: 2y² + 2xy = -x²
Dividing by 2: y² + xy = -½x²
Completing the square: y² + xy + ¼x² = -½x² + ¼x²
Factoring: (y + ½x)² = -¼x²
Solving: y + ½x = ±√ (-¼x²)
But the radical will be negative unless x = 0
If x = 0, y = 0.
When y = 0, x = 0 ---> So, it's only the origin, the point (0, 0).
CPhill: I think that you graphed 2x^2 + 4xy + 2y^2 = 0, not 2x^2 + 4xy + 4y^2 = 0
Looking at 2x² + 4xy + 4y² = 0
Divide by 2: x² + 2xy + 2y² = 0
Solving for y: 2y² + 2xy = -x²
Dividing by 2: y² + xy = -½x²
Completing the square: y² + xy + ¼x² = -½x² + ¼x²
Factoring: (y + ½x)² = -¼x²
Solving: y + ½x = ±√ (-¼x²)
But the radical will be negative unless x = 0
If x = 0, y = 0.
When y = 0, x = 0 ---> So, it's only the origin, the point (0, 0).