2x^2+xy-8y^2=0

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Solution for 2x^2+xy-8y^2=0 equation: 


Simplifying
2x2 + xy + -8y2 = 0

Reorder the terms:
xy + 2x2 + -8y2 = 0

Solving
xy + 2x2 + -8y2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
0.5xy + x2 + -4y2 = 0

Move the constant term to the right:

Add '4y2' to each side of the equation.
0.5xy + x2 + -4y2 + 4y2 = 0 + 4y2

Combine like terms: -4y2 + 4y2 = 0
0.5xy + x2 + 0 = 0 + 4y2
0.5xy + x2 = 0 + 4y2
Remove the zero:
0.5xy + x2 = 4y2

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.
0.5xy + x2 + 0.25y2 = 4y2 + 0.25y2

Combine like terms: 4y2 + 0.25y2 = 4.25y2
0.5xy + x2 + 0.25y2 = 4.25y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 4.25y2

Calculate the square root of the right side: 2.061552813y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 2.061552813y and -2.061552813y.


Subproblem 1
x + 0.5y = 2.061552813y

Simplifying
x + 0.5y = 2.061552813y

Solving
x + 0.5y = 2.061552813y

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 = 2.061552813y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 2.061552813y + -0.5y
x = 2.061552813y + -0.5y

Combine like terms: 2.061552813y + -0.5y = 1.561552813y
x = 1.561552813y

Simplifying
x = 1.561552813y


Subproblem 2
x + 0.5y = -2.061552813y

Simplifying
x + 0.5y = -2.061552813y

Solving
x + 0.5y = -2.061552813y

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 = -2.061552813y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -2.061552813y + -0.5y
x = -2.061552813y + -0.5y

Combine like terms: -2.061552813y + -0.5y = -2.561552813y
x = -2.561552813y

Simplifying
x = -2.561552813y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.561552813y, -2.561552813y}

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