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

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


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

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

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

Solving for variable 'x'.

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

Divide each side by '3'.
xy + x2 + -0.3333333333y2 = 0

Move the constant term to the right:

Add '0.3333333333y2' to each side of the equation.
xy + x2 + -0.3333333333y2 + 0.3333333333y2 = 0 + 0.3333333333y2

Combine like terms: -0.3333333333y2 + 0.3333333333y2 = 0.0000000000
xy + x2 + 0.0000000000 = 0 + 0.3333333333y2
xy + x2 = 0 + 0.3333333333y2
Remove the zero:
xy + x2 = 0.3333333333y2

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 = 0.3333333333y2 + 0.25y2

Reorder the terms:
x2 + y2 + 0.25y2 = 0.3333333333y2 + 0.25y2

Combine like terms: y2 + 0.25y2 = 1.25y2
x2 + 1.25y2 = 0.3333333333y2 + 0.25y2

Combine like terms: 0.3333333333y2 + 0.25y2 = 0.5833333333y2
x2 + 1.25y2 = 0.5833333333y2

Factor a perfect square on the left side:
(x + 0.5y)(x + 0.5y) = 0.5833333333y2

Calculate the square root of the right side: 0.763762616y

Break this problem into two subproblems by setting 
(x + 0.5y) equal to 0.763762616y and -0.763762616y.


Subproblem 1
x + 0.5y = 0.763762616y

Simplifying
x + 0.5y = 0.763762616y

Solving
x + 0.5y = 0.763762616y

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 = 0.763762616y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = 0.763762616y + -0.5y
x = 0.763762616y + -0.5y

Combine like terms: 0.763762616y + -0.5y = 0.263762616y
x = 0.263762616y

Simplifying
x = 0.263762616y


Subproblem 2
x + 0.5y = -0.763762616y

Simplifying
x + 0.5y = -0.763762616y

Solving
x + 0.5y = -0.763762616y

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 = -0.763762616y + -0.5y

Combine like terms: 0.5y + -0.5y = 0.0
x + 0.0 = -0.763762616y + -0.5y
x = -0.763762616y + -0.5y

Combine like terms: -0.763762616y + -0.5y = -1.263762616y
x = -1.263762616y

Simplifying
x = -1.263762616y


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.263762616y, -1.263762616y}

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