4166=6.28r^2+558.92r

Solution for 4166=6.28r^2+558.92r equation:

Simplifying
4166 = 6.28r2 + 558.92r

Reorder the terms:
4166 = 558.92r + 6.28r2

Solving
4166 = 558.92r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
4166 + -558.92r + -6.28r2 = 558.92r + -558.92r + 6.28r2 + -6.28r2

Combine like terms: 558.92r + -558.92r = 0.00
4166 + -558.92r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2
4166 + -558.92r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
4166 + -558.92r + -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'.
-663.3757962 + 89r + r2 = 0

Move the constant term to the right:

Add '663.3757962' to each side of the equation.
-663.3757962 + 89r + 663.3757962 + r2 = 0 + 663.3757962

Reorder the terms:
-663.3757962 + 663.3757962 + 89r + r2 = 0 + 663.3757962

Combine like terms: -663.3757962 + 663.3757962 = 0.0000000
0.0000000 + 89r + r2 = 0 + 663.3757962
89r + r2 = 0 + 663.3757962

Combine like terms: 0 + 663.3757962 = 663.3757962
89r + r2 = 663.3757962

The r term is 89r.  Take half its coefficient (44.5).
Square it (1980.25) and add it to both sides.

Add '1980.25' to each side of the equation.
89r + 1980.25 + r2 = 663.3757962 + 1980.25

Reorder the terms:
1980.25 + 89r + r2 = 663.3757962 + 1980.25

Combine like terms: 663.3757962 + 1980.25 = 2643.6257962
1980.25 + 89r + r2 = 2643.6257962

Factor a perfect square on the left side:
(r + 44.5)(r + 44.5) = 2643.6257962

Calculate the square root of the right side: 51.41620169

Break this problem into two subproblems by setting 
(r + 44.5) equal to 51.41620169 and -51.41620169.

Subproblem 1
r + 44.5 = 51.41620169

Simplifying
r + 44.5 = 51.41620169

Reorder the terms:
44.5 + r = 51.41620169

Solving
44.5 + r = 51.41620169

Solving for variable 'r'.

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

Add '-44.5' to each side of the equation.
44.5 + -44.5 + r = 51.41620169 + -44.5

Combine like terms: 44.5 + -44.5 = 0.0
0.0 + r = 51.41620169 + -44.5
r = 51.41620169 + -44.5

Combine like terms: 51.41620169 + -44.5 = 6.91620169
r = 6.91620169

Simplifying
r = 6.91620169

Subproblem 2
r + 44.5 = -51.41620169

Simplifying
r + 44.5 = -51.41620169

Reorder the terms:
44.5 + r = -51.41620169

Solving
44.5 + r = -51.41620169

Solving for variable 'r'.

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

Add '-44.5' to each side of the equation.
44.5 + -44.5 + r = -51.41620169 + -44.5

Combine like terms: 44.5 + -44.5 = 0.0
0.0 + r = -51.41620169 + -44.5
r = -51.41620169 + -44.5

Combine like terms: -51.41620169 + -44.5 = -95.91620169
r = -95.91620169

Simplifying
r = -95.91620169

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {6.91620169, -95.91620169}