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

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


Simplifying
3y2 + -1xy + -1x2 = 0

Reorder the terms:
-1xy + -1x2 + 3y2 = 0

Solving
-1xy + -1x2 + 3y2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
xy + x2 + -3y2 = 0

Move the constant term to the right:

Add '3y2' to each side of the equation.
xy + x2 + -3y2 + 3y2 = 0 + 3y2

Combine like terms: -3y2 + 3y2 = 0
xy + x2 + 0 = 0 + 3y2
xy + x2 = 0 + 3y2
Remove the zero:
xy + x2 = 3y2

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 = 3y2 + 0.25y2

Reorder the terms:
x2 + y2 + 0.25y2 = 3y2 + 0.25y2

Combine like terms: y2 + 0.25y2 = 1.25y2
x2 + 1.25y2 = 3y2 + 0.25y2

Combine like terms: 3y2 + 0.25y2 = 3.25y2
x2 + 1.25y2 = 3.25y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 3.25y2

Calculate the square root of the right side: 1.802775638y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 1.802775638y and -1.802775638y.


Subproblem 1
x + 0.5y = 1.802775638y

Simplifying
x + 0.5y = 1.802775638y

Solving
x + 0.5y = 1.802775638y

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 = 1.802775638y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 1.802775638y + -0.5y
x = 1.802775638y + -0.5y

Combine like terms: 1.802775638y + -0.5y = 1.302775638y
x = 1.302775638y

Simplifying
x = 1.302775638y


Subproblem 2
x + 0.5y = -1.802775638y

Simplifying
x + 0.5y = -1.802775638y

Solving
x + 0.5y = -1.802775638y

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 = -1.802775638y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -1.802775638y + -0.5y
x = -1.802775638y + -0.5y

Combine like terms: -1.802775638y + -0.5y = -2.302775638y
x = -2.302775638y

Simplifying
x = -2.302775638y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.302775638y, -2.302775638y}

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