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Simplifying 4(3 + -6d) + 5 = 9(2d + -2) * 3d (3 * 4 + -6d * 4) + 5 = 9(2d + -2) * 3d (12 + -24d) + 5 = 9(2d + -2) * 3d Reorder the terms: 12 + 5 + -24d = 9(2d + -2) * 3d Combine like terms: 12 + 5 = 17 17 + -24d = 9(2d + -2) * 3d Reorder the terms: 17 + -24d = 9(-2 + 2d) * 3d Reorder the terms for easier multiplication: 17 + -24d = 9 * 3d(-2 + 2d) Multiply 9 * 3 17 + -24d = 27d(-2 + 2d) 17 + -24d = (-2 * 27d + 2d * 27d) 17 + -24d = (-54d + 54d2) Solving 17 + -24d = -54d + 54d2 Solving for variable 'd'. Combine like terms: -24d + 54d = 30d 17 + 30d + -54d2 = -54d + 54d2 + 54d + -54d2 Reorder the terms: 17 + 30d + -54d2 = -54d + 54d + 54d2 + -54d2 Combine like terms: -54d + 54d = 0 17 + 30d + -54d2 = 0 + 54d2 + -54d2 17 + 30d + -54d2 = 54d2 + -54d2 Combine like terms: 54d2 + -54d2 = 0 17 + 30d + -54d2 = 0 Begin completing the square. Divide all terms by -54 the coefficient of the squared term: Divide each side by '-54'. -0.3148148148 + -0.5555555556d + d2 = 0 Move the constant term to the right: Add '0.3148148148' to each side of the equation. -0.3148148148 + -0.5555555556d + 0.3148148148 + d2 = 0 + 0.3148148148 Reorder the terms: -0.3148148148 + 0.3148148148 + -0.5555555556d + d2 = 0 + 0.3148148148 Combine like terms: -0.3148148148 + 0.3148148148 = 0.0000000000 0.0000000000 + -0.5555555556d + d2 = 0 + 0.3148148148 -0.5555555556d + d2 = 0 + 0.3148148148 Combine like terms: 0 + 0.3148148148 = 0.3148148148 -0.5555555556d + d2 = 0.3148148148 The d term is -0.5555555556d. Take half its coefficient (-0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. -0.5555555556d + 0.07716049384 + d2 = 0.3148148148 + 0.07716049384 Reorder the terms: 0.07716049384 + -0.5555555556d + d2 = 0.3148148148 + 0.07716049384 Combine like terms: 0.3148148148 + 0.07716049384 = 0.39197530864 0.07716049384 + -0.5555555556d + d2 = 0.39197530864 Factor a perfect square on the left side: (d + -0.2777777778)(d + -0.2777777778) = 0.39197530864 Calculate the square root of the right side: 0.626079315 Break this problem into two subproblems by setting (d + -0.2777777778) equal to 0.626079315 and -0.626079315.Subproblem 1
d + -0.2777777778 = 0.626079315 Simplifying d + -0.2777777778 = 0.626079315 Reorder the terms: -0.2777777778 + d = 0.626079315 Solving -0.2777777778 + d = 0.626079315 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + d = 0.626079315 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + d = 0.626079315 + 0.2777777778 d = 0.626079315 + 0.2777777778 Combine like terms: 0.626079315 + 0.2777777778 = 0.9038570928 d = 0.9038570928 Simplifying d = 0.9038570928Subproblem 2
d + -0.2777777778 = -0.626079315 Simplifying d + -0.2777777778 = -0.626079315 Reorder the terms: -0.2777777778 + d = -0.626079315 Solving -0.2777777778 + d = -0.626079315 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.2777777778' to each side of the equation. -0.2777777778 + 0.2777777778 + d = -0.626079315 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + d = -0.626079315 + 0.2777777778 d = -0.626079315 + 0.2777777778 Combine like terms: -0.626079315 + 0.2777777778 = -0.3483015372 d = -0.3483015372 Simplifying d = -0.3483015372Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.9038570928, -0.3483015372}
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