100=2d^2+6d+9

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Solution for 100=2d^2+6d+9 equation:


Simplifying
100 = 2d2 + 6d + 9

Reorder the terms:
100 = 9 + 6d + 2d2

Solving
100 = 9 + 6d + 2d2

Solving for variable 'd'.

Combine like terms: 100 + -9 = 91
91 + -6d + -2d2 = 9 + 6d + 2d2 + -9 + -6d + -2d2

Reorder the terms:
91 + -6d + -2d2 = 9 + -9 + 6d + -6d + 2d2 + -2d2

Combine like terms: 9 + -9 = 0
91 + -6d + -2d2 = 0 + 6d + -6d + 2d2 + -2d2
91 + -6d + -2d2 = 6d + -6d + 2d2 + -2d2

Combine like terms: 6d + -6d = 0
91 + -6d + -2d2 = 0 + 2d2 + -2d2
91 + -6d + -2d2 = 2d2 + -2d2

Combine like terms: 2d2 + -2d2 = 0
91 + -6d + -2d2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-45.5 + 3d + d2 = 0

Move the constant term to the right:

Add '45.5' to each side of the equation.
-45.5 + 3d + 45.5 + d2 = 0 + 45.5

Reorder the terms:
-45.5 + 45.5 + 3d + d2 = 0 + 45.5

Combine like terms: -45.5 + 45.5 = 0.0
0.0 + 3d + d2 = 0 + 45.5
3d + d2 = 0 + 45.5

Combine like terms: 0 + 45.5 = 45.5
3d + d2 = 45.5

The d term is 3d.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3d + 2.25 + d2 = 45.5 + 2.25

Reorder the terms:
2.25 + 3d + d2 = 45.5 + 2.25

Combine like terms: 45.5 + 2.25 = 47.75
2.25 + 3d + d2 = 47.75

Factor a perfect square on the left side:
(d + 1.5)(d + 1.5) = 47.75

Calculate the square root of the right side: 6.910137481

Break this problem into two subproblems by setting 
(d + 1.5) equal to 6.910137481 and -6.910137481.

Subproblem 1

d + 1.5 = 6.910137481 Simplifying d + 1.5 = 6.910137481 Reorder the terms: 1.5 + d = 6.910137481 Solving 1.5 + d = 6.910137481 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + d = 6.910137481 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + d = 6.910137481 + -1.5 d = 6.910137481 + -1.5 Combine like terms: 6.910137481 + -1.5 = 5.410137481 d = 5.410137481 Simplifying d = 5.410137481

Subproblem 2

d + 1.5 = -6.910137481 Simplifying d + 1.5 = -6.910137481 Reorder the terms: 1.5 + d = -6.910137481 Solving 1.5 + d = -6.910137481 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + d = -6.910137481 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + d = -6.910137481 + -1.5 d = -6.910137481 + -1.5 Combine like terms: -6.910137481 + -1.5 = -8.410137481 d = -8.410137481 Simplifying d = -8.410137481

Solution

The solution to the problem is based on the solutions from the subproblems. d = {5.410137481, -8.410137481}

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