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Simplifying 0 = 4d2 + 6d + -3 Reorder the terms: 0 = -3 + 6d + 4d2 Solving 0 = -3 + 6d + 4d2 Solving for variable 'd'. Combine like terms: 0 + 3 = 3 3 + -6d + -4d2 = -3 + 6d + 4d2 + 3 + -6d + -4d2 Reorder the terms: 3 + -6d + -4d2 = -3 + 3 + 6d + -6d + 4d2 + -4d2 Combine like terms: -3 + 3 = 0 3 + -6d + -4d2 = 0 + 6d + -6d + 4d2 + -4d2 3 + -6d + -4d2 = 6d + -6d + 4d2 + -4d2 Combine like terms: 6d + -6d = 0 3 + -6d + -4d2 = 0 + 4d2 + -4d2 3 + -6d + -4d2 = 4d2 + -4d2 Combine like terms: 4d2 + -4d2 = 0 3 + -6d + -4d2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.75 + 1.5d + d2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.5d + 0.75 + d2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.5d + d2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.5d + d2 = 0 + 0.75 1.5d + d2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.5d + d2 = 0.75 The d term is 1.5d. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5d + 0.5625 + d2 = 0.75 + 0.5625 Reorder the terms: 0.5625 + 1.5d + d2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + 1.5d + d2 = 1.3125 Factor a perfect square on the left side: (d + 0.75)(d + 0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (d + 0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
d + 0.75 = 1.145643924 Simplifying d + 0.75 = 1.145643924 Reorder the terms: 0.75 + d = 1.145643924 Solving 0.75 + d = 1.145643924 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + d = 1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + d = 1.145643924 + -0.75 d = 1.145643924 + -0.75 Combine like terms: 1.145643924 + -0.75 = 0.395643924 d = 0.395643924 Simplifying d = 0.395643924Subproblem 2
d + 0.75 = -1.145643924 Simplifying d + 0.75 = -1.145643924 Reorder the terms: 0.75 + d = -1.145643924 Solving 0.75 + d = -1.145643924 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + d = -1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + d = -1.145643924 + -0.75 d = -1.145643924 + -0.75 Combine like terms: -1.145643924 + -0.75 = -1.895643924 d = -1.895643924 Simplifying d = -1.895643924Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.395643924, -1.895643924}
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