(x+y)(x+y)=x+(x-y)(x-y)

Solution for (x+y)(x+y)=x+(x-y)(x-y) equation:

Simplifying
(x + y)(x + y) = x + (x + -1y)(x + -1y)

Multiply (x + y) * (x + y)
(x(x + y) + y(x + y)) = x + (x + -1y)(x + -1y)
((x * x + y * x) + y(x + y)) = x + (x + -1y)(x + -1y)

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

Reorder the terms:
(xy + xy + x2 + y2) = x + (x + -1y)(x + -1y)

Combine like terms: xy + xy = 2xy
(2xy + x2 + y2) = x + (x + -1y)(x + -1y)

Multiply (x + -1y) * (x + -1y)
2xy + x2 + y2 = x + (x(x + -1y) + -1y * (x + -1y))
2xy + x2 + y2 = x + ((x * x + -1y * x) + -1y * (x + -1y))

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

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

Combine like terms: -1xy + -1xy = -2xy
2xy + x2 + y2 = x + (-2xy + x2 + 1y2)

Add '-1x2' to each side of the equation.
2xy + x2 + -1x2 + y2 = x + -2xy + x2 + -1x2 + 1y2

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

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

Solving
2xy + y2 = x + -2xy + 1y2

Solving for variable 'x'.

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

Add '-1x' to each side of the equation.
2xy + -1x + y2 = x + -2xy + -1x + 1y2

Reorder the terms:
-1x + 2xy + y2 = x + -2xy + -1x + 1y2

Reorder the terms:
-1x + 2xy + y2 = x + -1x + -2xy + 1y2

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

Add '2xy' to each side of the equation.
-1x + 2xy + 2xy + y2 = -2xy + 2xy + 1y2

Combine like terms: 2xy + 2xy = 4xy
-1x + 4xy + y2 = -2xy + 2xy + 1y2

Combine like terms: -2xy + 2xy = 0
-1x + 4xy + y2 = 0 + 1y2
-1x + 4xy + y2 = 1y2

Add '-1y2' to each side of the equation.
-1x + 4xy + y2 + -1y2 = 1y2 + -1y2

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

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

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + 4y) = 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 '(-1 + 4y)' equal to zero and attempt to solve:

Simplifying
-1 + 4y = 0

Solving
-1 + 4y = 0

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

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

Combine like terms: -1 + 1 = 0
0 + 4y = 0 + 1
4y = 0 + 1

Combine like terms: 0 + 1 = 1
4y = 1

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

Combine like terms: 4y + -4y = 0
0 = 1 + -4y

Simplifying
0 = 1 + -4y

The solution to this equation could not be determined.

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