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Simplifying (6.41)(1.2 + x)(x) = (8 + -2x)(8 + -2x) Reorder the terms for easier multiplication: 6.41x(1.2 + x) = (8 + -2x)(8 + -2x) (1.2 * 6.41x + x * 6.41x) = (8 + -2x)(8 + -2x) (7.692x + 6.41x2) = (8 + -2x)(8 + -2x) Multiply (8 + -2x) * (8 + -2x) 7.692x + 6.41x2 = (8(8 + -2x) + -2x * (8 + -2x)) 7.692x + 6.41x2 = ((8 * 8 + -2x * 8) + -2x * (8 + -2x)) 7.692x + 6.41x2 = ((64 + -16x) + -2x * (8 + -2x)) 7.692x + 6.41x2 = (64 + -16x + (8 * -2x + -2x * -2x)) 7.692x + 6.41x2 = (64 + -16x + (-16x + 4x2)) Combine like terms: -16x + -16x = -32x 7.692x + 6.41x2 = (64 + -32x + 4x2) Solving 7.692x + 6.41x2 = 64 + -32x + 4x2 Solving for variable 'x'. Reorder the terms: -64 + 7.692x + 32x + 6.41x2 + -4x2 = 64 + -32x + 4x2 + -64 + 32x + -4x2 Combine like terms: 7.692x + 32x = 39.692x -64 + 39.692x + 6.41x2 + -4x2 = 64 + -32x + 4x2 + -64 + 32x + -4x2 Combine like terms: 6.41x2 + -4x2 = 2.41x2 -64 + 39.692x + 2.41x2 = 64 + -32x + 4x2 + -64 + 32x + -4x2 Reorder the terms: -64 + 39.692x + 2.41x2 = 64 + -64 + -32x + 32x + 4x2 + -4x2 Combine like terms: 64 + -64 = 0 -64 + 39.692x + 2.41x2 = 0 + -32x + 32x + 4x2 + -4x2 -64 + 39.692x + 2.41x2 = -32x + 32x + 4x2 + -4x2 Combine like terms: -32x + 32x = 0 -64 + 39.692x + 2.41x2 = 0 + 4x2 + -4x2 -64 + 39.692x + 2.41x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 -64 + 39.692x + 2.41x2 = 0 Begin completing the square. Divide all terms by 2.41 the coefficient of the squared term: Divide each side by '2.41'. -26.5560166 + 16.46970954x + x2 = 0 Move the constant term to the right: Add '26.5560166' to each side of the equation. -26.5560166 + 16.46970954x + 26.5560166 + x2 = 0 + 26.5560166 Reorder the terms: -26.5560166 + 26.5560166 + 16.46970954x + x2 = 0 + 26.5560166 Combine like terms: -26.5560166 + 26.5560166 = 0.0000000 0.0000000 + 16.46970954x + x2 = 0 + 26.5560166 16.46970954x + x2 = 0 + 26.5560166 Combine like terms: 0 + 26.5560166 = 26.5560166 16.46970954x + x2 = 26.5560166 The x term is 16.46970954x. Take half its coefficient (8.23485477). Square it (67.81283308) and add it to both sides. Add '67.81283308' to each side of the equation. 16.46970954x + 67.81283308 + x2 = 26.5560166 + 67.81283308 Reorder the terms: 67.81283308 + 16.46970954x + x2 = 26.5560166 + 67.81283308 Combine like terms: 26.5560166 + 67.81283308 = 94.36884968 67.81283308 + 16.46970954x + x2 = 94.36884968 Factor a perfect square on the left side: (x + 8.23485477)(x + 8.23485477) = 94.36884968 Calculate the square root of the right side: 9.714363061 Break this problem into two subproblems by setting (x + 8.23485477) equal to 9.714363061 and -9.714363061.Subproblem 1
x + 8.23485477 = 9.714363061 Simplifying x + 8.23485477 = 9.714363061 Reorder the terms: 8.23485477 + x = 9.714363061 Solving 8.23485477 + x = 9.714363061 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.23485477' to each side of the equation. 8.23485477 + -8.23485477 + x = 9.714363061 + -8.23485477 Combine like terms: 8.23485477 + -8.23485477 = 0.00000000 0.00000000 + x = 9.714363061 + -8.23485477 x = 9.714363061 + -8.23485477 Combine like terms: 9.714363061 + -8.23485477 = 1.479508291 x = 1.479508291 Simplifying x = 1.479508291Subproblem 2
x + 8.23485477 = -9.714363061 Simplifying x + 8.23485477 = -9.714363061 Reorder the terms: 8.23485477 + x = -9.714363061 Solving 8.23485477 + x = -9.714363061 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.23485477' to each side of the equation. 8.23485477 + -8.23485477 + x = -9.714363061 + -8.23485477 Combine like terms: 8.23485477 + -8.23485477 = 0.00000000 0.00000000 + x = -9.714363061 + -8.23485477 x = -9.714363061 + -8.23485477 Combine like terms: -9.714363061 + -8.23485477 = -17.949217831 x = -17.949217831 Simplifying x = -17.949217831Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.479508291, -17.949217831}
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