6d(d-1)=21-2

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Solution for 6d(d-1)=21-2 equation:


Simplifying
6d(d + -1) = 21 + -2

Reorder the terms:
6d(-1 + d) = 21 + -2
(-1 * 6d + d * 6d) = 21 + -2
(-6d + 6d2) = 21 + -2

Combine like terms: 21 + -2 = 19
-6d + 6d2 = 19

Solving
-6d + 6d2 = 19

Solving for variable 'd'.

Reorder the terms:
-19 + -6d + 6d2 = 19 + -19

Combine like terms: 19 + -19 = 0
-19 + -6d + 6d2 = 0

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

Divide each side by '6'.
-3.166666667 + -1d + d2 = 0

Move the constant term to the right:

Add '3.166666667' to each side of the equation.
-3.166666667 + -1d + 3.166666667 + d2 = 0 + 3.166666667

Reorder the terms:
-3.166666667 + 3.166666667 + -1d + d2 = 0 + 3.166666667

Combine like terms: -3.166666667 + 3.166666667 = 0.000000000
0.000000000 + -1d + d2 = 0 + 3.166666667
-1d + d2 = 0 + 3.166666667

Combine like terms: 0 + 3.166666667 = 3.166666667
-1d + d2 = 3.166666667

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 = 3.166666667 + 0.25

Reorder the terms:
0.25 + -1d + d2 = 3.166666667 + 0.25

Combine like terms: 3.166666667 + 0.25 = 3.416666667
0.25 + -1d + d2 = 3.416666667

Factor a perfect square on the left side:
(d + -0.5)(d + -0.5) = 3.416666667

Calculate the square root of the right side: 1.848422751

Break this problem into two subproblems by setting 
(d + -0.5) equal to 1.848422751 and -1.848422751.

Subproblem 1

d + -0.5 = 1.848422751 Simplifying d + -0.5 = 1.848422751 Reorder the terms: -0.5 + d = 1.848422751 Solving -0.5 + d = 1.848422751 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 = 1.848422751 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = 1.848422751 + 0.5 d = 1.848422751 + 0.5 Combine like terms: 1.848422751 + 0.5 = 2.348422751 d = 2.348422751 Simplifying d = 2.348422751

Subproblem 2

d + -0.5 = -1.848422751 Simplifying d + -0.5 = -1.848422751 Reorder the terms: -0.5 + d = -1.848422751 Solving -0.5 + d = -1.848422751 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 = -1.848422751 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + d = -1.848422751 + 0.5 d = -1.848422751 + 0.5 Combine like terms: -1.848422751 + 0.5 = -1.348422751 d = -1.348422751 Simplifying d = -1.348422751

Solution

The solution to the problem is based on the solutions from the subproblems. d = {2.348422751, -1.348422751}

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