(2x+3y)*dx+(2x+4y)*dy=0

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Solution for (2x+3y)*dx+(2x+4y)*dy=0 equation: 


Simplifying
(2x + 3y) * dx + (2x + 4y) * dy = 0

Reorder the terms for easier multiplication:
dx(2x + 3y) + (2x + 4y) * dy = 0
(2x * dx + 3y * dx) + (2x + 4y) * dy = 0

Reorder the terms:
(3dxy + 2dx2) + (2x + 4y) * dy = 0
(3dxy + 2dx2) + (2x + 4y) * dy = 0

Reorder the terms for easier multiplication:
3dxy + 2dx2 + dy(2x + 4y) = 0
3dxy + 2dx2 + (2x * dy + 4y * dy) = 0
3dxy + 2dx2 + (2dxy + 4dy2) = 0

Reorder the terms:
3dxy + 2dxy + 2dx2 + 4dy2 = 0

Combine like terms: 3dxy + 2dxy = 5dxy
5dxy + 2dx2 + 4dy2 = 0

Solving
5dxy + 2dx2 + 4dy2 = 0

Solving for variable 'd'.

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

Factor out the Greatest Common Factor (GCF), 'd'.
d(5xy + 2x2 + 4y2) = 0


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

Simplifying
d = 0

Solving
d = 0

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

Simplifying
d = 0

Subproblem 2
Set the factor '(5xy + 2x2 + 4y2)' equal to zero and attempt to solve:

Simplifying
5xy + 2x2 + 4y2 = 0

Solving
5xy + 2x2 + 4y2 = 0

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

Add '-5xy' to each side of the equation.
5xy + 2x2 + -5xy + 4y2 = 0 + -5xy

Reorder the terms:
5xy + -5xy + 2x2 + 4y2 = 0 + -5xy

Combine like terms: 5xy + -5xy = 0
0 + 2x2 + 4y2 = 0 + -5xy
2x2 + 4y2 = 0 + -5xy
Remove the zero:
2x2 + 4y2 = -5xy

Add '-2x2' to each side of the equation.
2x2 + -2x2 + 4y2 = -5xy + -2x2

Combine like terms: 2x2 + -2x2 = 0
0 + 4y2 = -5xy + -2x2
4y2 = -5xy + -2x2

Add '-4y2' to each side of the equation.
4y2 + -4y2 = -5xy + -2x2 + -4y2

Combine like terms: 4y2 + -4y2 = 0
0 = -5xy + -2x2 + -4y2

Simplifying
0 = -5xy + -2x2 + -4y2

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
d = {0}

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