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Simplifying (x + 3)(2 + x)(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 Reorder the terms: (3 + x)(2 + x)(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 Multiply (3 + x) * (2 + x) (3(2 + x) + x(2 + x))(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 ((2 * 3 + x * 3) + x(2 + x))(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 ((6 + 3x) + x(2 + x))(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 (6 + 3x + (2 * x + x * x))(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 (6 + 3x + (2x + x2))(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 Combine like terms: 3x + 2x = 5x (6 + 5x + x2)(3 + x) + (1 + -1x)(2 + -1x)(-1x) = 204 Multiply (6 + 5x + x2) * (3 + x) (6(3 + x) + 5x * (3 + x) + x2(3 + x)) + (1 + -1x)(2 + -1x)(-1x) = 204 ((3 * 6 + x * 6) + 5x * (3 + x) + x2(3 + x)) + (1 + -1x)(2 + -1x)(-1x) = 204 ((18 + 6x) + 5x * (3 + x) + x2(3 + x)) + (1 + -1x)(2 + -1x)(-1x) = 204 (18 + 6x + (3 * 5x + x * 5x) + x2(3 + x)) + (1 + -1x)(2 + -1x)(-1x) = 204 (18 + 6x + (15x + 5x2) + x2(3 + x)) + (1 + -1x)(2 + -1x)(-1x) = 204 (18 + 6x + 15x + 5x2 + (3 * x2 + x * x2)) + (1 + -1x)(2 + -1x)(-1x) = 204 (18 + 6x + 15x + 5x2 + (3x2 + x3)) + (1 + -1x)(2 + -1x)(-1x) = 204 Combine like terms: 6x + 15x = 21x (18 + 21x + 5x2 + 3x2 + x3) + (1 + -1x)(2 + -1x)(-1x) = 204 Combine like terms: 5x2 + 3x2 = 8x2 (18 + 21x + 8x2 + x3) + (1 + -1x)(2 + -1x)(-1x) = 204 Remove parenthesis around (-1x) 18 + 21x + 8x2 + x3 + (1 + -1x)(2 + -1x) * -1x = 204 Reorder the terms for easier multiplication: 18 + 21x + 8x2 + x3 + -1x(1 + -1x)(2 + -1x) = 204 Multiply (1 + -1x) * (2 + -1x) 18 + 21x + 8x2 + x3 + -1x(1(2 + -1x) + -1x * (2 + -1x)) = 204 18 + 21x + 8x2 + x3 + -1x((2 * 1 + -1x * 1) + -1x * (2 + -1x)) = 204 18 + 21x + 8x2 + x3 + -1x((2 + -1x) + -1x * (2 + -1x)) = 204 18 + 21x + 8x2 + x3 + -1x(2 + -1x + (2 * -1x + -1x * -1x)) = 204 18 + 21x + 8x2 + x3 + -1x(2 + -1x + (-2x + 1x2)) = 204 Combine like terms: -1x + -2x = -3x 18 + 21x + 8x2 + x3 + -1x(2 + -3x + 1x2) = 204 18 + 21x + 8x2 + x3 + (2 * -1x + -3x * -1x + 1x2 * -1x) = 204 18 + 21x + 8x2 + x3 + (-2x + 3x2 + -1x3) = 204 Reorder the terms: 18 + 21x + -2x + 8x2 + 3x2 + x3 + -1x3 = 204 Combine like terms: 21x + -2x = 19x 18 + 19x + 8x2 + 3x2 + x3 + -1x3 = 204 Combine like terms: 8x2 + 3x2 = 11x2 18 + 19x + 11x2 + x3 + -1x3 = 204 Combine like terms: x3 + -1x3 = 0 18 + 19x + 11x2 + 0 = 204 18 + 19x + 11x2 = 204 Solving 18 + 19x + 11x2 = 204 Solving for variable 'x'. Reorder the terms: 18 + -204 + 19x + 11x2 = 204 + -204 Combine like terms: 18 + -204 = -186 -186 + 19x + 11x2 = 204 + -204 Combine like terms: 204 + -204 = 0 -186 + 19x + 11x2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -16.90909091 + 1.727272727x + x2 = 0 Move the constant term to the right: Add '16.90909091' to each side of the equation. -16.90909091 + 1.727272727x + 16.90909091 + x2 = 0 + 16.90909091 Reorder the terms: -16.90909091 + 16.90909091 + 1.727272727x + x2 = 0 + 16.90909091 Combine like terms: -16.90909091 + 16.90909091 = 0.00000000 0.00000000 + 1.727272727x + x2 = 0 + 16.90909091 1.727272727x + x2 = 0 + 16.90909091 Combine like terms: 0 + 16.90909091 = 16.90909091 1.727272727x + x2 = 16.90909091 The x term is 1.727272727x. Take half its coefficient (0.8636363635). Square it (0.7458677684) and add it to both sides. Add '0.7458677684' to each side of the equation. 1.727272727x + 0.7458677684 + x2 = 16.90909091 + 0.7458677684 Reorder the terms: 0.7458677684 + 1.727272727x + x2 = 16.90909091 + 0.7458677684 Combine like terms: 16.90909091 + 0.7458677684 = 17.6549586784 0.7458677684 + 1.727272727x + x2 = 17.6549586784 Factor a perfect square on the left side: (x + 0.8636363635)(x + 0.8636363635) = 17.6549586784 Calculate the square root of the right side: 4.201780418 Break this problem into two subproblems by setting (x + 0.8636363635) equal to 4.201780418 and -4.201780418.Subproblem 1
x + 0.8636363635 = 4.201780418 Simplifying x + 0.8636363635 = 4.201780418 Reorder the terms: 0.8636363635 + x = 4.201780418 Solving 0.8636363635 + x = 4.201780418 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8636363635' to each side of the equation. 0.8636363635 + -0.8636363635 + x = 4.201780418 + -0.8636363635 Combine like terms: 0.8636363635 + -0.8636363635 = 0.0000000000 0.0000000000 + x = 4.201780418 + -0.8636363635 x = 4.201780418 + -0.8636363635 Combine like terms: 4.201780418 + -0.8636363635 = 3.3381440545 x = 3.3381440545 Simplifying x = 3.3381440545Subproblem 2
x + 0.8636363635 = -4.201780418 Simplifying x + 0.8636363635 = -4.201780418 Reorder the terms: 0.8636363635 + x = -4.201780418 Solving 0.8636363635 + x = -4.201780418 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.8636363635' to each side of the equation. 0.8636363635 + -0.8636363635 + x = -4.201780418 + -0.8636363635 Combine like terms: 0.8636363635 + -0.8636363635 = 0.0000000000 0.0000000000 + x = -4.201780418 + -0.8636363635 x = -4.201780418 + -0.8636363635 Combine like terms: -4.201780418 + -0.8636363635 = -5.0654167815 x = -5.0654167815 Simplifying x = -5.0654167815Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.3381440545, -5.0654167815}
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