y(y+2x)-x(y-3x)-(y^2-x^2)=0

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


Simplifying
y(y + 2x) + -1x(y + -3x) + -1(y2 + -1x2) = 0

Reorder the terms:
y(2x + y) + -1x(y + -3x) + -1(y2 + -1x2) = 0
(2x * y + y * y) + -1x(y + -3x) + -1(y2 + -1x2) = 0
(2xy + y2) + -1x(y + -3x) + -1(y2 + -1x2) = 0

Reorder the terms:
2xy + y2 + -1x(-3x + y) + -1(y2 + -1x2) = 0
2xy + y2 + (-3x * -1x + y * -1x) + -1(y2 + -1x2) = 0

Reorder the terms:
2xy + y2 + (-1xy + 3x2) + -1(y2 + -1x2) = 0
2xy + y2 + (-1xy + 3x2) + -1(y2 + -1x2) = 0

Reorder the terms:
2xy + y2 + -1xy + 3x2 + -1(-1x2 + y2) = 0
2xy + y2 + -1xy + 3x2 + (-1x2 * -1 + y2 * -1) = 0
2xy + y2 + -1xy + 3x2 + (1x2 + -1y2) = 0

Reorder the terms:
2xy + -1xy + 3x2 + 1x2 + y2 + -1y2 = 0

Combine like terms: 2xy + -1xy = 1xy
1xy + 3x2 + 1x2 + y2 + -1y2 = 0

Combine like terms: 3x2 + 1x2 = 4x2
1xy + 4x2 + y2 + -1y2 = 0

Combine like terms: y2 + -1y2 = 0
1xy + 4x2 + 0 = 0
1xy + 4x2 = 0

Solving
1xy + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(y + 4x) = 0


Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

Move all terms containing x to the left, all other terms to the right.

Simplifying
x = 0

Subproblem 2
Set the factor '(y + 4x)' equal to zero and attempt to solve:

Simplifying
y + 4x = 0

Reorder the terms:
4x + y = 0

Solving
4x + y = 0

Move all terms containing x to the left, all other terms to the right.

Add '-1y' to each side of the equation.
4x + y + -1y = 0 + -1y

Combine like terms: y + -1y = 0
4x + 0 = 0 + -1y
4x = 0 + -1y
Remove the zero:
4x = -1y

Divide each side by '4'.
x = -0.25y

Simplifying
x = -0.25y
Solution
x = {0, -0.25y}

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