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Simplifying 3d + -4d(d + -12) = 16 Reorder the terms: 3d + -4d(-12 + d) = 16 3d + (-12 * -4d + d * -4d) = 16 3d + (48d + -4d2) = 16 Combine like terms: 3d + 48d = 51d 51d + -4d2 = 16 Solving 51d + -4d2 = 16 Solving for variable 'd'. Reorder the terms: -16 + 51d + -4d2 = 16 + -16 Combine like terms: 16 + -16 = 0 -16 + 51d + -4d2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 4 + -12.75d + d2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -12.75d + -4 + d2 = 0 + -4 Reorder the terms: 4 + -4 + -12.75d + d2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -12.75d + d2 = 0 + -4 -12.75d + d2 = 0 + -4 Combine like terms: 0 + -4 = -4 -12.75d + d2 = -4 The d term is -12.75d. Take half its coefficient (-6.375). Square it (40.640625) and add it to both sides. Add '40.640625' to each side of the equation. -12.75d + 40.640625 + d2 = -4 + 40.640625 Reorder the terms: 40.640625 + -12.75d + d2 = -4 + 40.640625 Combine like terms: -4 + 40.640625 = 36.640625 40.640625 + -12.75d + d2 = 36.640625 Factor a perfect square on the left side: (d + -6.375)(d + -6.375) = 36.640625 Calculate the square root of the right side: 6.053150006 Break this problem into two subproblems by setting (d + -6.375) equal to 6.053150006 and -6.053150006.Subproblem 1
d + -6.375 = 6.053150006 Simplifying d + -6.375 = 6.053150006 Reorder the terms: -6.375 + d = 6.053150006 Solving -6.375 + d = 6.053150006 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '6.375' to each side of the equation. -6.375 + 6.375 + d = 6.053150006 + 6.375 Combine like terms: -6.375 + 6.375 = 0.000 0.000 + d = 6.053150006 + 6.375 d = 6.053150006 + 6.375 Combine like terms: 6.053150006 + 6.375 = 12.428150006 d = 12.428150006 Simplifying d = 12.428150006Subproblem 2
d + -6.375 = -6.053150006 Simplifying d + -6.375 = -6.053150006 Reorder the terms: -6.375 + d = -6.053150006 Solving -6.375 + d = -6.053150006 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '6.375' to each side of the equation. -6.375 + 6.375 + d = -6.053150006 + 6.375 Combine like terms: -6.375 + 6.375 = 0.000 0.000 + d = -6.053150006 + 6.375 d = -6.053150006 + 6.375 Combine like terms: -6.053150006 + 6.375 = 0.321849994 d = 0.321849994 Simplifying d = 0.321849994Solution
The solution to the problem is based on the solutions from the subproblems. d = {12.428150006, 0.321849994}
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