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Simplifying ((x + y)(x + -1y) + -1(x + y)(x + y))(2(x + y)) = 0 Multiply (x + y) * (x + -1y) ((x(x + -1y) + y(x + -1y)) + -1(x + y)(x + y))(2(x + y)) = 0 (((x * x + -1y * x) + y(x + -1y)) + -1(x + y)(x + y))(2(x + y)) = 0 Reorder the terms: (((-1xy + x2) + y(x + -1y)) + -1(x + y)(x + y))(2(x + y)) = 0 (((-1xy + x2) + y(x + -1y)) + -1(x + y)(x + y))(2(x + y)) = 0 ((-1xy + x2 + (x * y + -1y * y)) + -1(x + y)(x + y))(2(x + y)) = 0 ((-1xy + x2 + (xy + -1y2)) + -1(x + y)(x + y))(2(x + y)) = 0 Reorder the terms: ((-1xy + xy + x2 + -1y2) + -1(x + y)(x + y))(2(x + y)) = 0 Combine like terms: -1xy + xy = 0 ((0 + x2 + -1y2) + -1(x + y)(x + y))(2(x + y)) = 0 ((x2 + -1y2) + -1(x + y)(x + y))(2(x + y)) = 0 Multiply (x + y) * (x + y) (x2 + -1y2 + -1(x(x + y) + y(x + y)))(2(x + y)) = 0 (x2 + -1y2 + -1((x * x + y * x) + y(x + y)))(2(x + y)) = 0 Reorder the terms: (x2 + -1y2 + -1((xy + x2) + y(x + y)))(2(x + y)) = 0 (x2 + -1y2 + -1((xy + x2) + y(x + y)))(2(x + y)) = 0 (x2 + -1y2 + -1(xy + x2 + (x * y + y * y)))(2(x + y)) = 0 (x2 + -1y2 + -1(xy + x2 + (xy + y2)))(2(x + y)) = 0 Reorder the terms: (x2 + -1y2 + -1(xy + xy + x2 + y2))(2(x + y)) = 0 Combine like terms: xy + xy = 2xy (x2 + -1y2 + -1(2xy + x2 + y2))(2(x + y)) = 0 (x2 + -1y2 + (2xy * -1 + x2 * -1 + y2 * -1))(2(x + y)) = 0 (x2 + -1y2 + (-2xy + -1x2 + -1y2))(2(x + y)) = 0 Reorder the terms: (-2xy + x2 + -1x2 + -1y2 + -1y2)(2(x + y)) = 0 Combine like terms: x2 + -1x2 = 0 (-2xy + 0 + -1y2 + -1y2)(2(x + y)) = 0 (-2xy + -1y2 + -1y2)(2(x + y)) = 0 Combine like terms: -1y2 + -1y2 = -2y2 (-2xy + -2y2)(2(x + y)) = 0 (-2xy + -2y2)((x * 2 + y * 2)) = 0 (-2xy + -2y2)((2x + 2y)) = 0 Multiply (-2xy + -2y2) * (2x + 2y) (-2xy * (2x + 2y) + -2y2 * (2x + 2y)) = 0 ((2x * -2xy + 2y * -2xy) + -2y2 * (2x + 2y)) = 0 Reorder the terms: ((-4xy2 + -4x2y) + -2y2 * (2x + 2y)) = 0 ((-4xy2 + -4x2y) + -2y2 * (2x + 2y)) = 0 (-4xy2 + -4x2y + (2x * -2y2 + 2y * -2y2)) = 0 (-4xy2 + -4x2y + (-4xy2 + -4y3)) = 0 Reorder the terms: (-4xy2 + -4xy2 + -4x2y + -4y3) = 0 Combine like terms: -4xy2 + -4xy2 = -8xy2 (-8xy2 + -4x2y + -4y3) = 0 Solving -8xy2 + -4x2y + -4y3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '-4y'. -4y(2xy + x2 + y2) = 0 Factor a trinomial. -4y((x + y)(x + y)) = 0 Ignore the factor -4.Subproblem 1
Set the factor 'y' equal to zero and attempt to solve: Simplifying y = 0 Solving y = 0 Move all terms containing x to the left, all other terms to the right. Add '-1y' to each side of the equation. y + -1y = 0 + -1y Remove the zero: 0 = -1y Simplifying 0 = -1y The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(x + y)' equal to zero and attempt to solve: Simplifying x + y = 0 Solving x + y = 0 Move all terms containing x to the left, all other terms to the right. Add '-1y' to each side of the equation. x + y + -1y = 0 + -1y Combine like terms: y + -1y = 0 x + 0 = 0 + -1y x = 0 + -1y Remove the zero: x = -1y Simplifying x = -1ySubproblem 3
Set the factor '(x + y)' equal to zero and attempt to solve: Simplifying x + y = 0 Solving x + y = 0 Move all terms containing x to the left, all other terms to the right. Add '-1y' to each side of the equation. x + y + -1y = 0 + -1y Combine like terms: y + -1y = 0 x + 0 = 0 + -1y x = 0 + -1y Remove the zero: x = -1y Simplifying x = -1ySolution
x = {-1y, -1y}
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