406.8184=2*3.14*r(r)+2*3.14*r*11.7

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Solution for 406.8184=2*3.14*r(r)+2*3.14*r*11.7 equation: 


Simplifying
406.8184 = 2 * 3.14r(r) + 2 * 3.14r * 11.7

Multiply 2 * 3.14
406.8184 = 6.28r * r + 2 * 3.14r * 11.7

Multiply r * r
406.8184 = 6.28r2 + 2 * 3.14r * 11.7

Reorder the terms for easier multiplication:
406.8184 = 6.28r2 + 2 * 3.14 * 11.7r

Multiply 2 * 3.14
406.8184 = 6.28r2 + 6.28 * 11.7r

Multiply 6.28 * 11.7
406.8184 = 6.28r2 + 73.476r

Reorder the terms:
406.8184 = 73.476r + 6.28r2

Solving
406.8184 = 73.476r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
406.8184 + -73.476r + -6.28r2 = 73.476r + -73.476r + 6.28r2 + -6.28r2

Combine like terms: 73.476r + -73.476r = 0.000
406.8184 + -73.476r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
406.8184 + -73.476r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
406.8184 + -73.476r + -6.28r2 = 0.00

Begin completing the square.  Divide all terms by
-6.28 the coefficient of the squared term: 

Divide each side by '-6.28'.
-64.78 + 11.7r + r2 = 0

Move the constant term to the right:

Add '64.78' to each side of the equation.
-64.78 + 11.7r + 64.78 + r2 = 0 + 64.78

Reorder the terms:
-64.78 + 64.78 + 11.7r + r2 = 0 + 64.78

Combine like terms: -64.78 + 64.78 = 0.00
0.00 + 11.7r + r2 = 0 + 64.78
11.7r + r2 = 0 + 64.78

Combine like terms: 0 + 64.78 = 64.78
11.7r + r2 = 64.78

The r term is 11.7r.  Take half its coefficient (5.85).
Square it (34.2225) and add it to both sides.

Add '34.2225' to each side of the equation.
11.7r + 34.2225 + r2 = 64.78 + 34.2225

Reorder the terms:
34.2225 + 11.7r + r2 = 64.78 + 34.2225

Combine like terms: 64.78 + 34.2225 = 99.0025
34.2225 + 11.7r + r2 = 99.0025

Factor a perfect square on the left side:
(r + 5.85)(r + 5.85) = 99.0025

Calculate the square root of the right side: 9.95

Break this problem into two subproblems by setting 
(r + 5.85) equal to 9.95 and -9.95.


Subproblem 1
r + 5.85 = 9.95

Simplifying
r + 5.85 = 9.95

Reorder the terms:
5.85 + r = 9.95

Solving
5.85 + r = 9.95

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-5.85' to each side of the equation.
5.85 + -5.85 + r = 9.95 + -5.85

Combine like terms: 5.85 + -5.85 = 0.00
0.00 + r = 9.95 + -5.85
r = 9.95 + -5.85

Combine like terms: 9.95 + -5.85 = 4.1
r = 4.1

Simplifying
r = 4.1


Subproblem 2
r + 5.85 = -9.95

Simplifying
r + 5.85 = -9.95

Reorder the terms:
5.85 + r = -9.95

Solving
5.85 + r = -9.95

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-5.85' to each side of the equation.
5.85 + -5.85 + r = -9.95 + -5.85

Combine like terms: 5.85 + -5.85 = 0.00
0.00 + r = -9.95 + -5.85
r = -9.95 + -5.85

Combine like terms: -9.95 + -5.85 = -15.8
r = -15.8

Simplifying
r = -15.8


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.1, -15.8}

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