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Simplifying 6d + 3(1 + d)(d + -7) = 2d Reorder the terms: 6d + 3(1 + d)(-7 + d) = 2d Multiply (1 + d) * (-7 + d) 6d + 3(1(-7 + d) + d(-7 + d)) = 2d 6d + 3((-7 * 1 + d * 1) + d(-7 + d)) = 2d 6d + 3((-7 + 1d) + d(-7 + d)) = 2d 6d + 3(-7 + 1d + (-7 * d + d * d)) = 2d 6d + 3(-7 + 1d + (-7d + d2)) = 2d Combine like terms: 1d + -7d = -6d 6d + 3(-7 + -6d + d2) = 2d 6d + (-7 * 3 + -6d * 3 + d2 * 3) = 2d 6d + (-21 + -18d + 3d2) = 2d Reorder the terms: -21 + 6d + -18d + 3d2 = 2d Combine like terms: 6d + -18d = -12d -21 + -12d + 3d2 = 2d Solving -21 + -12d + 3d2 = 2d Solving for variable 'd'. Reorder the terms: -21 + -12d + -2d + 3d2 = 2d + -2d Combine like terms: -12d + -2d = -14d -21 + -14d + 3d2 = 2d + -2d Combine like terms: 2d + -2d = 0 -21 + -14d + 3d2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -7 + -4.666666667d + d2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + -4.666666667d + 7 + d2 = 0 + 7 Reorder the terms: -7 + 7 + -4.666666667d + d2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -4.666666667d + d2 = 0 + 7 -4.666666667d + d2 = 0 + 7 Combine like terms: 0 + 7 = 7 -4.666666667d + d2 = 7 The d term is -4.666666667d. Take half its coefficient (-2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. -4.666666667d + 5.444444448 + d2 = 7 + 5.444444448 Reorder the terms: 5.444444448 + -4.666666667d + d2 = 7 + 5.444444448 Combine like terms: 7 + 5.444444448 = 12.444444448 5.444444448 + -4.666666667d + d2 = 12.444444448 Factor a perfect square on the left side: (d + -2.333333334)(d + -2.333333334) = 12.444444448 Calculate the square root of the right side: 3.527668415 Break this problem into two subproblems by setting (d + -2.333333334) equal to 3.527668415 and -3.527668415.Subproblem 1
d + -2.333333334 = 3.527668415 Simplifying d + -2.333333334 = 3.527668415 Reorder the terms: -2.333333334 + d = 3.527668415 Solving -2.333333334 + d = 3.527668415 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + d = 3.527668415 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + d = 3.527668415 + 2.333333334 d = 3.527668415 + 2.333333334 Combine like terms: 3.527668415 + 2.333333334 = 5.861001749 d = 5.861001749 Simplifying d = 5.861001749Subproblem 2
d + -2.333333334 = -3.527668415 Simplifying d + -2.333333334 = -3.527668415 Reorder the terms: -2.333333334 + d = -3.527668415 Solving -2.333333334 + d = -3.527668415 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '2.333333334' to each side of the equation. -2.333333334 + 2.333333334 + d = -3.527668415 + 2.333333334 Combine like terms: -2.333333334 + 2.333333334 = 0.000000000 0.000000000 + d = -3.527668415 + 2.333333334 d = -3.527668415 + 2.333333334 Combine like terms: -3.527668415 + 2.333333334 = -1.194335081 d = -1.194335081 Simplifying d = -1.194335081Solution
The solution to the problem is based on the solutions from the subproblems. d = {5.861001749, -1.194335081}
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