8x^2+xy-3y^2=0

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Solution for 8x^2+xy-3y^2=0 equation:


Simplifying
8x2 + xy + -3y2 = 0

Reorder the terms:
xy + 8x2 + -3y2 = 0

Solving
xy + 8x2 + -3y2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
0.125xy + x2 + -0.375y2 = 0

Move the constant term to the right:

Add '0.375y2' to each side of the equation.
0.125xy + x2 + -0.375y2 + 0.375y2 = 0 + 0.375y2

Combine like terms: -0.375y2 + 0.375y2 = 0.000
0.125xy + x2 + 0.000 = 0 + 0.375y2
0.125xy + x2 = 0 + 0.375y2
Remove the zero:
0.125xy + x2 = 0.375y2

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.
0.125xy + x2 + 0.25y2 = 0.375y2 + 0.25y2

Combine like terms: 0.375y2 + 0.25y2 = 0.625y2
0.125xy + x2 + 0.25y2 = 0.625y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 0.625y2

Calculate the square root of the right side: 0.790569415y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 0.790569415y and -0.790569415y.

Subproblem 1

x + 0.5y = 0.790569415y Simplifying x + 0.5y = 0.790569415y Solving x + 0.5y = 0.790569415y 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 = 0.790569415y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = 0.790569415y + -0.5y x = 0.790569415y + -0.5y Combine like terms: 0.790569415y + -0.5y = 0.290569415y x = 0.290569415y Simplifying x = 0.290569415y

Subproblem 2

x + 0.5y = -0.790569415y Simplifying x + 0.5y = -0.790569415y Solving x + 0.5y = -0.790569415y 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 = -0.790569415y + -0.5y Combine like terms: 0.5y + -0.5y = 0.0 x + 0.0 = -0.790569415y + -0.5y x = -0.790569415y + -0.5y Combine like terms: -0.790569415y + -0.5y = -1.290569415y x = -1.290569415y Simplifying x = -1.290569415y

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.290569415y, -1.290569415y}

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