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Simplifying (1 + -1d)(d + -1) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Reorder the terms: (1 + -1d)(-1 + d) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Multiply (1 + -1d) * (-1 + d) (1(-1 + d) + -1d * (-1 + d)) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) ((-1 * 1 + d * 1) + -1d * (-1 + d)) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) ((-1 + 1d) + -1d * (-1 + d)) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) (-1 + 1d + (-1 * -1d + d * -1d)) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) (-1 + 1d + (1d + -1d2)) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: 1d + 1d = 2d (-1 + 2d + -1d2) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -1 + 2d + -1d2 + (1 * d + d * d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -1 + 2d + -1d2 + (1d + d2) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -1 + 2d + -1d2 + 1d + d2 + (1 * -3 + -1d * -3) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -1 + 2d + -1d2 + 1d + d2 + (-3 + 3d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Reorder the terms: -1 + 2d + -1d2 + 1d + d2 + -3 + 3d + 5(-1 + d) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -1 + 2d + -1d2 + 1d + d2 + -3 + 3d + (-1 * 5 + d * 5) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -1 + 2d + -1d2 + 1d + d2 + -3 + 3d + (-5 + 5d) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Reorder the terms: -1 + -3 + -5 + 2d + 1d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: -1 + -3 = -4 -4 + -5 + 2d + 1d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: -4 + -5 = -9 -9 + 2d + 1d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: 2d + 1d = 3d -9 + 3d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: 3d + 3d = 6d -9 + 6d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: 6d + 5d = 11d -9 + 11d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: -1d2 + d2 = 0 -9 + 11d + 0 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) -9 + 11d = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Reorder the terms: -9 + 11d = (1 + d)(d + 2) + -2(1 + d)(2 + d) + -1d(d + -1) Reorder the terms: -9 + 11d = (1 + d)(2 + d) + -2(1 + d)(2 + d) + -1d(d + -1) Multiply (1 + d) * (2 + d) -9 + 11d = (1(2 + d) + d(2 + d)) + -2(1 + d)(2 + d) + -1d(d + -1) -9 + 11d = ((2 * 1 + d * 1) + d(2 + d)) + -2(1 + d)(2 + d) + -1d(d + -1) -9 + 11d = ((2 + 1d) + d(2 + d)) + -2(1 + d)(2 + d) + -1d(d + -1) -9 + 11d = (2 + 1d + (2 * d + d * d)) + -2(1 + d)(2 + d) + -1d(d + -1) -9 + 11d = (2 + 1d + (2d + d2)) + -2(1 + d)(2 + d) + -1d(d + -1) Combine like terms: 1d + 2d = 3d -9 + 11d = (2 + 3d + d2) + -2(1 + d)(2 + d) + -1d(d + -1) Multiply (1 + d) * (2 + d) -9 + 11d = 2 + 3d + d2 + -2(1(2 + d) + d(2 + d)) + -1d(d + -1) -9 + 11d = 2 + 3d + d2 + -2((2 * 1 + d * 1) + d(2 + d)) + -1d(d + -1) -9 + 11d = 2 + 3d + d2 + -2((2 + 1d) + d(2 + d)) + -1d(d + -1) -9 + 11d = 2 + 3d + d2 + -2(2 + 1d + (2 * d + d * d)) + -1d(d + -1) -9 + 11d = 2 + 3d + d2 + -2(2 + 1d + (2d + d2)) + -1d(d + -1) Combine like terms: 1d + 2d = 3d -9 + 11d = 2 + 3d + d2 + -2(2 + 3d + d2) + -1d(d + -1) -9 + 11d = 2 + 3d + d2 + (2 * -2 + 3d * -2 + d2 * -2) + -1d(d + -1) -9 + 11d = 2 + 3d + d2 + (-4 + -6d + -2d2) + -1d(d + -1) Reorder the terms: -9 + 11d = 2 + 3d + d2 + -4 + -6d + -2d2 + -1d(-1 + d) -9 + 11d = 2 + 3d + d2 + -4 + -6d + -2d2 + (-1 * -1d + d * -1d) -9 + 11d = 2 + 3d + d2 + -4 + -6d + -2d2 + (1d + -1d2) Reorder the terms: -9 + 11d = 2 + -4 + 3d + -6d + 1d + d2 + -2d2 + -1d2 Combine like terms: 2 + -4 = -2 -9 + 11d = -2 + 3d + -6d + 1d + d2 + -2d2 + -1d2 Combine like terms: 3d + -6d = -3d -9 + 11d = -2 + -3d + 1d + d2 + -2d2 + -1d2 Combine like terms: -3d + 1d = -2d -9 + 11d = -2 + -2d + d2 + -2d2 + -1d2 Combine like terms: d2 + -2d2 = -1d2 -9 + 11d = -2 + -2d + -1d2 + -1d2 Combine like terms: -1d2 + -1d2 = -2d2 -9 + 11d = -2 + -2d + -2d2 Solving -9 + 11d = -2 + -2d + -2d2 Solving for variable 'd'. Reorder the terms: -9 + 2 + 11d + 2d + 2d2 = -2 + -2d + -2d2 + 2 + 2d + 2d2 Combine like terms: -9 + 2 = -7 -7 + 11d + 2d + 2d2 = -2 + -2d + -2d2 + 2 + 2d + 2d2 Combine like terms: 11d + 2d = 13d -7 + 13d + 2d2 = -2 + -2d + -2d2 + 2 + 2d + 2d2 Reorder the terms: -7 + 13d + 2d2 = -2 + 2 + -2d + 2d + -2d2 + 2d2 Combine like terms: -2 + 2 = 0 -7 + 13d + 2d2 = 0 + -2d + 2d + -2d2 + 2d2 -7 + 13d + 2d2 = -2d + 2d + -2d2 + 2d2 Combine like terms: -2d + 2d = 0 -7 + 13d + 2d2 = 0 + -2d2 + 2d2 -7 + 13d + 2d2 = -2d2 + 2d2 Combine like terms: -2d2 + 2d2 = 0 -7 + 13d + 2d2 = 0 Factor a trinomial. (-7 + -1d)(1 + -2d) = 0Subproblem 1
Set the factor '(-7 + -1d)' equal to zero and attempt to solve: Simplifying -7 + -1d = 0 Solving -7 + -1d = 0 Move all terms containing d to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + -1d = 0 + 7 Combine like terms: -7 + 7 = 0 0 + -1d = 0 + 7 -1d = 0 + 7 Combine like terms: 0 + 7 = 7 -1d = 7 Divide each side by '-1'. d = -7 Simplifying d = -7Subproblem 2
Set the factor '(1 + -2d)' equal to zero and attempt to solve: Simplifying 1 + -2d = 0 Solving 1 + -2d = 0 Move all terms containing d to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + -2d = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2d = 0 + -1 -2d = 0 + -1 Combine like terms: 0 + -1 = -1 -2d = -1 Divide each side by '-2'. d = 0.5 Simplifying d = 0.5Solution
d = {-7, 0.5}
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