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Simplifying (x + 2011)(x + 2013)(x + 2014) = (x + 2013)(x + 2014)(x + 2015) Reorder the terms: (2011 + x)(x + 2013)(x + 2014) = (x + 2013)(x + 2014)(x + 2015) Reorder the terms: (2011 + x)(2013 + x)(x + 2014) = (x + 2013)(x + 2014)(x + 2015) Reorder the terms: (2011 + x)(2013 + x)(2014 + x) = (x + 2013)(x + 2014)(x + 2015) Multiply (2011 + x) * (2013 + x) (2011(2013 + x) + x(2013 + x))(2014 + x) = (x + 2013)(x + 2014)(x + 2015) ((2013 * 2011 + x * 2011) + x(2013 + x))(2014 + x) = (x + 2013)(x + 2014)(x + 2015) ((4048143 + 2011x) + x(2013 + x))(2014 + x) = (x + 2013)(x + 2014)(x + 2015) (4048143 + 2011x + (2013 * x + x * x))(2014 + x) = (x + 2013)(x + 2014)(x + 2015) (4048143 + 2011x + (2013x + x2))(2014 + x) = (x + 2013)(x + 2014)(x + 2015) Combine like terms: 2011x + 2013x = 4024x (4048143 + 4024x + x2)(2014 + x) = (x + 2013)(x + 2014)(x + 2015) Multiply (4048143 + 4024x + x2) * (2014 + x) (4048143(2014 + x) + 4024x * (2014 + x) + x2(2014 + x)) = (x + 2013)(x + 2014)(x + 2015) ((2014 * 4048143 + x * 4048143) + 4024x * (2014 + x) + x2(2014 + x)) = (x + 2013)(x + 2014)(x + 2015) ((8152960002 + 4048143x) + 4024x * (2014 + x) + x2(2014 + x)) = (x + 2013)(x + 2014)(x + 2015) (8152960002 + 4048143x + (2014 * 4024x + x * 4024x) + x2(2014 + x)) = (x + 2013)(x + 2014)(x + 2015) (8152960002 + 4048143x + (8104336x + 4024x2) + x2(2014 + x)) = (x + 2013)(x + 2014)(x + 2015) (8152960002 + 4048143x + 8104336x + 4024x2 + (2014 * x2 + x * x2)) = (x + 2013)(x + 2014)(x + 2015) (8152960002 + 4048143x + 8104336x + 4024x2 + (2014x2 + x3)) = (x + 2013)(x + 2014)(x + 2015) Combine like terms: 4048143x + 8104336x = 12152479x (8152960002 + 12152479x + 4024x2 + 2014x2 + x3) = (x + 2013)(x + 2014)(x + 2015) Combine like terms: 4024x2 + 2014x2 = 6038x2 (8152960002 + 12152479x + 6038x2 + x3) = (x + 2013)(x + 2014)(x + 2015) Reorder the terms: 8152960002 + 12152479x + 6038x2 + x3 = (2013 + x)(x + 2014)(x + 2015) Reorder the terms: 8152960002 + 12152479x + 6038x2 + x3 = (2013 + x)(2014 + x)(x + 2015) Reorder the terms: 8152960002 + 12152479x + 6038x2 + x3 = (2013 + x)(2014 + x)(2015 + x) Multiply (2013 + x) * (2014 + x) 8152960002 + 12152479x + 6038x2 + x3 = (2013(2014 + x) + x(2014 + x))(2015 + x) 8152960002 + 12152479x + 6038x2 + x3 = ((2014 * 2013 + x * 2013) + x(2014 + x))(2015 + x) 8152960002 + 12152479x + 6038x2 + x3 = ((4054182 + 2013x) + x(2014 + x))(2015 + x) 8152960002 + 12152479x + 6038x2 + x3 = (4054182 + 2013x + (2014 * x + x * x))(2015 + x) 8152960002 + 12152479x + 6038x2 + x3 = (4054182 + 2013x + (2014x + x2))(2015 + x) Combine like terms: 2013x + 2014x = 4027x 8152960002 + 12152479x + 6038x2 + x3 = (4054182 + 4027x + x2)(2015 + x) Multiply (4054182 + 4027x + x2) * (2015 + x) 8152960002 + 12152479x + 6038x2 + x3 = (4054182(2015 + x) + 4027x * (2015 + x) + x2(2015 + x)) 8152960002 + 12152479x + 6038x2 + x3 = ((2015 * 4054182 + x * 4054182) + 4027x * (2015 + x) + x2(2015 + x)) 8152960002 + 12152479x + 6038x2 + x3 = ((8169176730 + 4054182x) + 4027x * (2015 + x) + x2(2015 + x)) 8152960002 + 12152479x + 6038x2 + x3 = (8169176730 + 4054182x + (2015 * 4027x + x * 4027x) + x2(2015 + x)) 8152960002 + 12152479x + 6038x2 + x3 = (8169176730 + 4054182x + (8114405x + 4027x2) + x2(2015 + x)) 8152960002 + 12152479x + 6038x2 + x3 = (8169176730 + 4054182x + 8114405x + 4027x2 + (2015 * x2 + x * x2)) 8152960002 + 12152479x + 6038x2 + x3 = (8169176730 + 4054182x + 8114405x + 4027x2 + (2015x2 + x3)) Combine like terms: 4054182x + 8114405x = 12168587x 8152960002 + 12152479x + 6038x2 + x3 = (8169176730 + 12168587x + 4027x2 + 2015x2 + x3) Combine like terms: 4027x2 + 2015x2 = 6042x2 8152960002 + 12152479x + 6038x2 + x3 = (8169176730 + 12168587x + 6042x2 + x3) Add '-1x3' to each side of the equation. 8152960002 + 12152479x + 6038x2 + x3 + -1x3 = 8169176730 + 12168587x + 6042x2 + x3 + -1x3 Combine like terms: x3 + -1x3 = 0 8152960002 + 12152479x + 6038x2 + 0 = 8169176730 + 12168587x + 6042x2 + x3 + -1x3 8152960002 + 12152479x + 6038x2 = 8169176730 + 12168587x + 6042x2 + x3 + -1x3 Combine like terms: x3 + -1x3 = 0 8152960002 + 12152479x + 6038x2 = 8169176730 + 12168587x + 6042x2 + 0 8152960002 + 12152479x + 6038x2 = 8169176730 + 12168587x + 6042x2 Solving 8152960002 + 12152479x + 6038x2 = 8169176730 + 12168587x + 6042x2 Solving for variable 'x'. Reorder the terms: 8152960002 + -8169176730 + 12152479x + -12168587x + 6038x2 + -6042x2 = 8169176730 + 12168587x + 6042x2 + -8169176730 + -12168587x + -6042x2 Combine like terms: 8152960002 + -8169176730 = -16216728 -16216728 + 12152479x + -12168587x + 6038x2 + -6042x2 = 8169176730 + 12168587x + 6042x2 + -8169176730 + -12168587x + -6042x2 Combine like terms: 12152479x + -12168587x = -16108x -16216728 + -16108x + 6038x2 + -6042x2 = 8169176730 + 12168587x + 6042x2 + -8169176730 + -12168587x + -6042x2 Combine like terms: 6038x2 + -6042x2 = -4x2 -16216728 + -16108x + -4x2 = 8169176730 + 12168587x + 6042x2 + -8169176730 + -12168587x + -6042x2 Reorder the terms: -16216728 + -16108x + -4x2 = 8169176730 + -8169176730 + 12168587x + -12168587x + 6042x2 + -6042x2 Combine like terms: 8169176730 + -8169176730 = 0 -16216728 + -16108x + -4x2 = 0 + 12168587x + -12168587x + 6042x2 + -6042x2 -16216728 + -16108x + -4x2 = 12168587x + -12168587x + 6042x2 + -6042x2 Combine like terms: 12168587x + -12168587x = 0 -16216728 + -16108x + -4x2 = 0 + 6042x2 + -6042x2 -16216728 + -16108x + -4x2 = 6042x2 + -6042x2 Combine like terms: 6042x2 + -6042x2 = 0 -16216728 + -16108x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '-4'. -4(4054182 + 4027x + x2) = 0 Factor a trinomial. -4((2014 + x)(2013 + x)) = 0 Ignore the factor -4.Subproblem 1
Set the factor '(2014 + x)' equal to zero and attempt to solve: Simplifying 2014 + x = 0 Solving 2014 + x = 0 Move all terms containing x to the left, all other terms to the right. Add '-2014' to each side of the equation. 2014 + -2014 + x = 0 + -2014 Combine like terms: 2014 + -2014 = 0 0 + x = 0 + -2014 x = 0 + -2014 Combine like terms: 0 + -2014 = -2014 x = -2014 Simplifying x = -2014Subproblem 2
Set the factor '(2013 + x)' equal to zero and attempt to solve: Simplifying 2013 + x = 0 Solving 2013 + x = 0 Move all terms containing x to the left, all other terms to the right. Add '-2013' to each side of the equation. 2013 + -2013 + x = 0 + -2013 Combine like terms: 2013 + -2013 = 0 0 + x = 0 + -2013 x = 0 + -2013 Combine like terms: 0 + -2013 = -2013 x = -2013 Simplifying x = -2013Solution
x = {-2014, -2013}
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