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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 = -7 -7 + 24d = 9(2d + -2) * 3d Reorder the terms: -7 + 24d = 9(-2 + 2d) * 3d Reorder the terms for easier multiplication: -7 + 24d = 9 * 3d(-2 + 2d) Multiply 9 * 3 -7 + 24d = 27d(-2 + 2d) -7 + 24d = (-2 * 27d + 2d * 27d) -7 + 24d = (-54d + 54d2) Solving -7 + 24d = -54d + 54d2 Solving for variable 'd'. Combine like terms: 24d + 54d = 78d -7 + 78d + -54d2 = -54d + 54d2 + 54d + -54d2 Reorder the terms: -7 + 78d + -54d2 = -54d + 54d + 54d2 + -54d2 Combine like terms: -54d + 54d = 0 -7 + 78d + -54d2 = 0 + 54d2 + -54d2 -7 + 78d + -54d2 = 54d2 + -54d2 Combine like terms: 54d2 + -54d2 = 0 -7 + 78d + -54d2 = 0 Begin completing the square. Divide all terms by -54 the coefficient of the squared term: Divide each side by '-54'. 0.1296296296 + -1.444444444d + d2 = 0 Move the constant term to the right: Add '-0.1296296296' to each side of the equation. 0.1296296296 + -1.444444444d + -0.1296296296 + d2 = 0 + -0.1296296296 Reorder the terms: 0.1296296296 + -0.1296296296 + -1.444444444d + d2 = 0 + -0.1296296296 Combine like terms: 0.1296296296 + -0.1296296296 = 0.0000000000 0.0000000000 + -1.444444444d + d2 = 0 + -0.1296296296 -1.444444444d + d2 = 0 + -0.1296296296 Combine like terms: 0 + -0.1296296296 = -0.1296296296 -1.444444444d + d2 = -0.1296296296 The d term is -1.444444444d. Take half its coefficient (-0.722222222). Square it (0.5216049380) and add it to both sides. Add '0.5216049380' to each side of the equation. -1.444444444d + 0.5216049380 + d2 = -0.1296296296 + 0.5216049380 Reorder the terms: 0.5216049380 + -1.444444444d + d2 = -0.1296296296 + 0.5216049380 Combine like terms: -0.1296296296 + 0.5216049380 = 0.3919753084 0.5216049380 + -1.444444444d + d2 = 0.3919753084 Factor a perfect square on the left side: (d + -0.722222222)(d + -0.722222222) = 0.3919753084 Calculate the square root of the right side: 0.626079315 Break this problem into two subproblems by setting (d + -0.722222222) equal to 0.626079315 and -0.626079315.Subproblem 1
d + -0.722222222 = 0.626079315 Simplifying d + -0.722222222 = 0.626079315 Reorder the terms: -0.722222222 + d = 0.626079315 Solving -0.722222222 + d = 0.626079315 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.722222222' to each side of the equation. -0.722222222 + 0.722222222 + d = 0.626079315 + 0.722222222 Combine like terms: -0.722222222 + 0.722222222 = 0.000000000 0.000000000 + d = 0.626079315 + 0.722222222 d = 0.626079315 + 0.722222222 Combine like terms: 0.626079315 + 0.722222222 = 1.348301537 d = 1.348301537 Simplifying d = 1.348301537Subproblem 2
d + -0.722222222 = -0.626079315 Simplifying d + -0.722222222 = -0.626079315 Reorder the terms: -0.722222222 + d = -0.626079315 Solving -0.722222222 + d = -0.626079315 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.722222222' to each side of the equation. -0.722222222 + 0.722222222 + d = -0.626079315 + 0.722222222 Combine like terms: -0.722222222 + 0.722222222 = 0.000000000 0.000000000 + d = -0.626079315 + 0.722222222 d = -0.626079315 + 0.722222222 Combine like terms: -0.626079315 + 0.722222222 = 0.096142907 d = 0.096142907 Simplifying d = 0.096142907Solution
The solution to the problem is based on the solutions from the subproblems. d = {1.348301537, 0.096142907}
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