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Simplifying -9d + -24 = -3(d + 8) * 6d Reorder the terms: -24 + -9d = -3(d + 8) * 6d Reorder the terms: -24 + -9d = -3(8 + d) * 6d Reorder the terms for easier multiplication: -24 + -9d = -3 * 6d(8 + d) Multiply -3 * 6 -24 + -9d = -18d(8 + d) -24 + -9d = (8 * -18d + d * -18d) -24 + -9d = (-144d + -18d2) Solving -24 + -9d = -144d + -18d2 Solving for variable 'd'. Combine like terms: -9d + 144d = 135d -24 + 135d + 18d2 = -144d + -18d2 + 144d + 18d2 Reorder the terms: -24 + 135d + 18d2 = -144d + 144d + -18d2 + 18d2 Combine like terms: -144d + 144d = 0 -24 + 135d + 18d2 = 0 + -18d2 + 18d2 -24 + 135d + 18d2 = -18d2 + 18d2 Combine like terms: -18d2 + 18d2 = 0 -24 + 135d + 18d2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-8 + 45d + 6d2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-8 + 45d + 6d2)' equal to zero and attempt to solve: Simplifying -8 + 45d + 6d2 = 0 Solving -8 + 45d + 6d2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + 7.5d + d2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 7.5d + 1.333333333 + d2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 7.5d + d2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 7.5d + d2 = 0 + 1.333333333 7.5d + d2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 7.5d + d2 = 1.333333333 The d term is 7.5d. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5d + 14.0625 + d2 = 1.333333333 + 14.0625 Reorder the terms: 14.0625 + 7.5d + d2 = 1.333333333 + 14.0625 Combine like terms: 1.333333333 + 14.0625 = 15.395833333 14.0625 + 7.5d + d2 = 15.395833333 Factor a perfect square on the left side: (d + 3.75)(d + 3.75) = 15.395833333 Calculate the square root of the right side: 3.923752456 Break this problem into two subproblems by setting (d + 3.75) equal to 3.923752456 and -3.923752456.Subproblem 1
d + 3.75 = 3.923752456 Simplifying d + 3.75 = 3.923752456 Reorder the terms: 3.75 + d = 3.923752456 Solving 3.75 + d = 3.923752456 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + d = 3.923752456 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + d = 3.923752456 + -3.75 d = 3.923752456 + -3.75 Combine like terms: 3.923752456 + -3.75 = 0.173752456 d = 0.173752456 Simplifying d = 0.173752456Subproblem 2
d + 3.75 = -3.923752456 Simplifying d + 3.75 = -3.923752456 Reorder the terms: 3.75 + d = -3.923752456 Solving 3.75 + d = -3.923752456 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + d = -3.923752456 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + d = -3.923752456 + -3.75 d = -3.923752456 + -3.75 Combine like terms: -3.923752456 + -3.75 = -7.673752456 d = -7.673752456 Simplifying d = -7.673752456Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.173752456, -7.673752456}Solution
d = {0.173752456, -7.673752456}
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