d(d+1)=2(d+4)

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Solution for d(d+1)=2(d+4) equation:


Simplifying
d(d + 1) = 2(d + 4)

Reorder the terms:
d(1 + d) = 2(d + 4)
(1 * d + d * d) = 2(d + 4)
(1d + d2) = 2(d + 4)

Reorder the terms:
1d + d2 = 2(4 + d)
1d + d2 = (4 * 2 + d * 2)
1d + d2 = (8 + 2d)

Solving
1d + d2 = 8 + 2d

Solving for variable 'd'.

Reorder the terms:
-8 + 1d + -2d + d2 = 8 + 2d + -8 + -2d

Combine like terms: 1d + -2d = -1d
-8 + -1d + d2 = 8 + 2d + -8 + -2d

Reorder the terms:
-8 + -1d + d2 = 8 + -8 + 2d + -2d

Combine like terms: 8 + -8 = 0
-8 + -1d + d2 = 0 + 2d + -2d
-8 + -1d + d2 = 2d + -2d

Combine like terms: 2d + -2d = 0
-8 + -1d + d2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + -1d + 8 + d2 = 0 + 8

Reorder the terms:
-8 + 8 + -1d + d2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + -1d + d2 = 0 + 8
-1d + d2 = 0 + 8

Combine like terms: 0 + 8 = 8
-1d + d2 = 8

The d term is -1d.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1d + 0.25 + d2 = 8 + 0.25

Reorder the terms:
0.25 + -1d + d2 = 8 + 0.25

Combine like terms: 8 + 0.25 = 8.25
0.25 + -1d + d2 = 8.25

Factor a perfect square on the left side:
(d + -0.5)(d + -0.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(d + -0.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

d + -0.5 = 2.872281323 Simplifying d + -0.5 = 2.872281323 Reorder the terms: -0.5 + d = 2.872281323 Solving -0.5 + d = 2.872281323 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + d = 2.872281323 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = 2.872281323 + 0.5 d = 2.872281323 + 0.5 Combine like terms: 2.872281323 + 0.5 = 3.372281323 d = 3.372281323 Simplifying d = 3.372281323

Subproblem 2

d + -0.5 = -2.872281323 Simplifying d + -0.5 = -2.872281323 Reorder the terms: -0.5 + d = -2.872281323 Solving -0.5 + d = -2.872281323 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + d = -2.872281323 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = -2.872281323 + 0.5 d = -2.872281323 + 0.5 Combine like terms: -2.872281323 + 0.5 = -2.372281323 d = -2.372281323 Simplifying d = -2.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. d = {3.372281323, -2.372281323}

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