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Simplifying 4x + 5 + 4x + 3 = 3(4x + 3)(4x + 5) Reorder the terms: 5 + 3 + 4x + 4x = 3(4x + 3)(4x + 5) Combine like terms: 5 + 3 = 8 8 + 4x + 4x = 3(4x + 3)(4x + 5) Combine like terms: 4x + 4x = 8x 8 + 8x = 3(4x + 3)(4x + 5) Reorder the terms: 8 + 8x = 3(3 + 4x)(4x + 5) Reorder the terms: 8 + 8x = 3(3 + 4x)(5 + 4x) Multiply (3 + 4x) * (5 + 4x) 8 + 8x = 3(3(5 + 4x) + 4x * (5 + 4x)) 8 + 8x = 3((5 * 3 + 4x * 3) + 4x * (5 + 4x)) 8 + 8x = 3((15 + 12x) + 4x * (5 + 4x)) 8 + 8x = 3(15 + 12x + (5 * 4x + 4x * 4x)) 8 + 8x = 3(15 + 12x + (20x + 16x2)) Combine like terms: 12x + 20x = 32x 8 + 8x = 3(15 + 32x + 16x2) 8 + 8x = (15 * 3 + 32x * 3 + 16x2 * 3) 8 + 8x = (45 + 96x + 48x2) Solving 8 + 8x = 45 + 96x + 48x2 Solving for variable 'x'. Reorder the terms: 8 + -45 + 8x + -96x + -48x2 = 45 + 96x + 48x2 + -45 + -96x + -48x2 Combine like terms: 8 + -45 = -37 -37 + 8x + -96x + -48x2 = 45 + 96x + 48x2 + -45 + -96x + -48x2 Combine like terms: 8x + -96x = -88x -37 + -88x + -48x2 = 45 + 96x + 48x2 + -45 + -96x + -48x2 Reorder the terms: -37 + -88x + -48x2 = 45 + -45 + 96x + -96x + 48x2 + -48x2 Combine like terms: 45 + -45 = 0 -37 + -88x + -48x2 = 0 + 96x + -96x + 48x2 + -48x2 -37 + -88x + -48x2 = 96x + -96x + 48x2 + -48x2 Combine like terms: 96x + -96x = 0 -37 + -88x + -48x2 = 0 + 48x2 + -48x2 -37 + -88x + -48x2 = 48x2 + -48x2 Combine like terms: 48x2 + -48x2 = 0 -37 + -88x + -48x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(37 + 88x + 48x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(37 + 88x + 48x2)' equal to zero and attempt to solve: Simplifying 37 + 88x + 48x2 = 0 Solving 37 + 88x + 48x2 = 0 Begin completing the square. Divide all terms by 48 the coefficient of the squared term: Divide each side by '48'. 0.7708333333 + 1.833333333x + x2 = 0 Move the constant term to the right: Add '-0.7708333333' to each side of the equation. 0.7708333333 + 1.833333333x + -0.7708333333 + x2 = 0 + -0.7708333333 Reorder the terms: 0.7708333333 + -0.7708333333 + 1.833333333x + x2 = 0 + -0.7708333333 Combine like terms: 0.7708333333 + -0.7708333333 = 0.0000000000 0.0000000000 + 1.833333333x + x2 = 0 + -0.7708333333 1.833333333x + x2 = 0 + -0.7708333333 Combine like terms: 0 + -0.7708333333 = -0.7708333333 1.833333333x + x2 = -0.7708333333 The x term is 1.833333333x. Take half its coefficient (0.9166666665). Square it (0.8402777775) and add it to both sides. Add '0.8402777775' to each side of the equation. 1.833333333x + 0.8402777775 + x2 = -0.7708333333 + 0.8402777775 Reorder the terms: 0.8402777775 + 1.833333333x + x2 = -0.7708333333 + 0.8402777775 Combine like terms: -0.7708333333 + 0.8402777775 = 0.0694444442 0.8402777775 + 1.833333333x + x2 = 0.0694444442 Factor a perfect square on the left side: (x + 0.9166666665)(x + 0.9166666665) = 0.0694444442 Calculate the square root of the right side: 0.263523138 Break this problem into two subproblems by setting (x + 0.9166666665) equal to 0.263523138 and -0.263523138.Subproblem 1
x + 0.9166666665 = 0.263523138 Simplifying x + 0.9166666665 = 0.263523138 Reorder the terms: 0.9166666665 + x = 0.263523138 Solving 0.9166666665 + x = 0.263523138 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = 0.263523138 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = 0.263523138 + -0.9166666665 x = 0.263523138 + -0.9166666665 Combine like terms: 0.263523138 + -0.9166666665 = -0.6531435285 x = -0.6531435285 Simplifying x = -0.6531435285Subproblem 2
x + 0.9166666665 = -0.263523138 Simplifying x + 0.9166666665 = -0.263523138 Reorder the terms: 0.9166666665 + x = -0.263523138 Solving 0.9166666665 + x = -0.263523138 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.9166666665' to each side of the equation. 0.9166666665 + -0.9166666665 + x = -0.263523138 + -0.9166666665 Combine like terms: 0.9166666665 + -0.9166666665 = 0.0000000000 0.0000000000 + x = -0.263523138 + -0.9166666665 x = -0.263523138 + -0.9166666665 Combine like terms: -0.263523138 + -0.9166666665 = -1.1801898045 x = -1.1801898045 Simplifying x = -1.1801898045Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.6531435285, -1.1801898045}Solution
x = {-0.6531435285, -1.1801898045}
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