(1-d)(d-1)+d(1+d)-3(1-d)+5(d-1)=(d+1)(d+2)-2(1+d)(2+d)-2d(d-1)

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Solution for (1-d)(d-1)+d(1+d)-3(1-d)+5(d-1)=(d+1)(d+2)-2(1+d)(2+d)-2d(d-1) equation:


Simplifying
(1 + -1d)(d + -1) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(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) + -2d(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) + -2d(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) + -2d(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) + -2d(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) + -2d(d + -1)
(-1 + 1d + (1d + -1d2)) + d(1 + d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(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) + -2d(d + -1)
-1 + 2d + -1d2 + (1 * d + d * d) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)
-1 + 2d + -1d2 + (1d + d2) + -3(1 + -1d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)
-1 + 2d + -1d2 + 1d + d2 + (1 * -3 + -1d * -3) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)
-1 + 2d + -1d2 + 1d + d2 + (-3 + 3d) + 5(d + -1) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Reorder the terms:
-1 + 2d + -1d2 + 1d + d2 + -3 + 3d + 5(-1 + d) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)
-1 + 2d + -1d2 + 1d + d2 + -3 + 3d + (-1 * 5 + d * 5) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)
-1 + 2d + -1d2 + 1d + d2 + -3 + 3d + (-5 + 5d) = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Reorder the terms:
-1 + -3 + -5 + 2d + 1d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(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) + -2d(d + -1)

Combine like terms: -4 + -5 = -9
-9 + 2d + 1d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Combine like terms: 2d + 1d = 3d
-9 + 3d + 3d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Combine like terms: 3d + 3d = 6d
-9 + 6d + 5d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Combine like terms: 6d + 5d = 11d
-9 + 11d + -1d2 + d2 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Combine like terms: -1d2 + d2 = 0
-9 + 11d + 0 = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)
-9 + 11d = (d + 1)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Reorder the terms:
-9 + 11d = (1 + d)(d + 2) + -2(1 + d)(2 + d) + -2d(d + -1)

Reorder the terms:
-9 + 11d = (1 + d)(2 + d) + -2(1 + d)(2 + d) + -2d(d + -1)

Multiply (1 + d) * (2 + d)
-9 + 11d = (1(2 + d) + d(2 + d)) + -2(1 + d)(2 + d) + -2d(d + -1)
-9 + 11d = ((2 * 1 + d * 1) + d(2 + d)) + -2(1 + d)(2 + d) + -2d(d + -1)
-9 + 11d = ((2 + 1d) + d(2 + d)) + -2(1 + d)(2 + d) + -2d(d + -1)
-9 + 11d = (2 + 1d + (2 * d + d * d)) + -2(1 + d)(2 + d) + -2d(d + -1)
-9 + 11d = (2 + 1d + (2d + d2)) + -2(1 + d)(2 + d) + -2d(d + -1)

Combine like terms: 1d + 2d = 3d
-9 + 11d = (2 + 3d + d2) + -2(1 + d)(2 + d) + -2d(d + -1)

Multiply (1 + d) * (2 + d)
-9 + 11d = 2 + 3d + d2 + -2(1(2 + d) + d(2 + d)) + -2d(d + -1)
-9 + 11d = 2 + 3d + d2 + -2((2 * 1 + d * 1) + d(2 + d)) + -2d(d + -1)
-9 + 11d = 2 + 3d + d2 + -2((2 + 1d) + d(2 + d)) + -2d(d + -1)
-9 + 11d = 2 + 3d + d2 + -2(2 + 1d + (2 * d + d * d)) + -2d(d + -1)
-9 + 11d = 2 + 3d + d2 + -2(2 + 1d + (2d + d2)) + -2d(d + -1)

Combine like terms: 1d + 2d = 3d
-9 + 11d = 2 + 3d + d2 + -2(2 + 3d + d2) + -2d(d + -1)
-9 + 11d = 2 + 3d + d2 + (2 * -2 + 3d * -2 + d2 * -2) + -2d(d + -1)
-9 + 11d = 2 + 3d + d2 + (-4 + -6d + -2d2) + -2d(d + -1)

Reorder the terms:
-9 + 11d = 2 + 3d + d2 + -4 + -6d + -2d2 + -2d(-1 + d)
-9 + 11d = 2 + 3d + d2 + -4 + -6d + -2d2 + (-1 * -2d + d * -2d)
-9 + 11d = 2 + 3d + d2 + -4 + -6d + -2d2 + (2d + -2d2)

Reorder the terms:
-9 + 11d = 2 + -4 + 3d + -6d + 2d + d2 + -2d2 + -2d2

Combine like terms: 2 + -4 = -2
-9 + 11d = -2 + 3d + -6d + 2d + d2 + -2d2 + -2d2

Combine like terms: 3d + -6d = -3d
-9 + 11d = -2 + -3d + 2d + d2 + -2d2 + -2d2

Combine like terms: -3d + 2d = -1d
-9 + 11d = -2 + -1d + d2 + -2d2 + -2d2

Combine like terms: d2 + -2d2 = -1d2
-9 + 11d = -2 + -1d + -1d2 + -2d2

Combine like terms: -1d2 + -2d2 = -3d2
-9 + 11d = -2 + -1d + -3d2

Solving
-9 + 11d = -2 + -1d + -3d2

Solving for variable 'd'.

Reorder the terms:
-9 + 2 + 11d + d + 3d2 = -2 + -1d + -3d2 + 2 + d + 3d2

Combine like terms: -9 + 2 = -7
-7 + 11d + d + 3d2 = -2 + -1d + -3d2 + 2 + d + 3d2

Combine like terms: 11d + d = 12d
-7 + 12d + 3d2 = -2 + -1d + -3d2 + 2 + d + 3d2

Reorder the terms:
-7 + 12d + 3d2 = -2 + 2 + -1d + d + -3d2 + 3d2

Combine like terms: -2 + 2 = 0
-7 + 12d + 3d2 = 0 + -1d + d + -3d2 + 3d2
-7 + 12d + 3d2 = -1d + d + -3d2 + 3d2

Combine like terms: -1d + d = 0
-7 + 12d + 3d2 = 0 + -3d2 + 3d2
-7 + 12d + 3d2 = -3d2 + 3d2

Combine like terms: -3d2 + 3d2 = 0
-7 + 12d + 3d2 = 0

Begin completing the square.  Divide all terms by
3 the coefficient of the squared term: 

Divide each side by '3'.
-2.333333333 + 4d + d2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + 4d + 2.333333333 + d2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + 4d + d2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + 4d + d2 = 0 + 2.333333333
4d + d2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
4d + d2 = 2.333333333

The d term is 4d.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4d + 4 + d2 = 2.333333333 + 4

Reorder the terms:
4 + 4d + d2 = 2.333333333 + 4

Combine like terms: 2.333333333 + 4 = 6.333333333
4 + 4d + d2 = 6.333333333

Factor a perfect square on the left side:
(d + 2)(d + 2) = 6.333333333

Calculate the square root of the right side: 2.516611478

Break this problem into two subproblems by setting 
(d + 2) equal to 2.516611478 and -2.516611478.

Subproblem 1

d + 2 = 2.516611478 Simplifying d + 2 = 2.516611478 Reorder the terms: 2 + d = 2.516611478 Solving 2 + d = 2.516611478 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + d = 2.516611478 + -2 Combine like terms: 2 + -2 = 0 0 + d = 2.516611478 + -2 d = 2.516611478 + -2 Combine like terms: 2.516611478 + -2 = 0.516611478 d = 0.516611478 Simplifying d = 0.516611478

Subproblem 2

d + 2 = -2.516611478 Simplifying d + 2 = -2.516611478 Reorder the terms: 2 + d = -2.516611478 Solving 2 + d = -2.516611478 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + d = -2.516611478 + -2 Combine like terms: 2 + -2 = 0 0 + d = -2.516611478 + -2 d = -2.516611478 + -2 Combine like terms: -2.516611478 + -2 = -4.516611478 d = -4.516611478 Simplifying d = -4.516611478

Solution

The solution to the problem is based on the solutions from the subproblems. d = {0.516611478, -4.516611478}

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