4x^2+4xy-y^2=0

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Solution for 4x^2+4xy-y^2=0 equation:


Simplifying
4x2 + 4xy + -1y2 = 0

Reorder the terms:
4xy + 4x2 + -1y2 = 0

Solving
4xy + 4x2 + -1y2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
xy + x2 + -0.25y2 = 0

Move the constant term to the right:

Add '0.25y2' to each side of the equation.
xy + x2 + -0.25y2 + 0.25y2 = 0 + 0.25y2

Combine like terms: -0.25y2 + 0.25y2 = 0.00
xy + x2 + 0.00 = 0 + 0.25y2
xy + x2 = 0 + 0.25y2
Remove the zero:
xy + x2 = 0.25y2

The x term is xy.  Take half its coefficient (0.5y).
Square it (0.25y2) and add it to both sides.

Add '0.25y2' to each side of the equation.
y2 + x2 + 0.25y2 = 0.25y2 + 0.25y2

Reorder the terms:
x2 + y2 + 0.25y2 = 0.25y2 + 0.25y2

Combine like terms: y2 + 0.25y2 = 1.25y2
x2 + 1.25y2 = 0.25y2 + 0.25y2

Combine like terms: 0.25y2 + 0.25y2 = 0.5y2
x2 + 1.25y2 = 0.5y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 0.5y2

Calculate the square root of the right side: 0.707106781y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 0.707106781y and -0.707106781y.

Subproblem 1

x + 0.5y = 0.707106781y Simplifying x + 0.5y = 0.707106781y Solving x + 0.5y = 0.707106781y Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5y' to each side of the equation. x + 0.5y + -0.5y = 0.707106781y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = 0.707106781y + -0.5y x = 0.707106781y + -0.5y Combine like terms: 0.707106781y + -0.5y = 0.207106781y x = 0.207106781y Simplifying x = 0.207106781y

Subproblem 2

x + 0.5y = -0.707106781y Simplifying x + 0.5y = -0.707106781y Solving x + 0.5y = -0.707106781y Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5y' to each side of the equation. x + 0.5y + -0.5y = -0.707106781y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = -0.707106781y + -0.5y x = -0.707106781y + -0.5y Combine like terms: -0.707106781y + -0.5y = -1.207106781y x = -1.207106781y Simplifying x = -1.207106781y

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.207106781y, -1.207106781y}

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