0=6.28r^2+1456.96r-47533.32

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Solution for 0=6.28r^2+1456.96r-47533.32 equation: 


Simplifying
0 = 6.28r2 + 1456.96r + -47533.32

Reorder the terms:
0 = -47533.32 + 1456.96r + 6.28r2

Solving
0 = -47533.32 + 1456.96r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 47533.32 = 47533.32
47533.32 + -1456.96r + -6.28r2 = -47533.32 + 1456.96r + 6.28r2 + 47533.32 + -1456.96r + -6.28r2

Reorder the terms:
47533.32 + -1456.96r + -6.28r2 = -47533.32 + 47533.32 + 1456.96r + -1456.96r + 6.28r2 + -6.28r2

Combine like terms: -47533.32 + 47533.32 = 0.00
47533.32 + -1456.96r + -6.28r2 = 0.00 + 1456.96r + -1456.96r + 6.28r2 + -6.28r2
47533.32 + -1456.96r + -6.28r2 = 1456.96r + -1456.96r + 6.28r2 + -6.28r2

Combine like terms: 1456.96r + -1456.96r = 0.00
47533.32 + -1456.96r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2
47533.32 + -1456.96r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
47533.32 + -1456.96r + -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'.
-7569 + 232r + r2 = 0

Move the constant term to the right:

Add '7569' to each side of the equation.
-7569 + 232r + 7569 + r2 = 0 + 7569

Reorder the terms:
-7569 + 7569 + 232r + r2 = 0 + 7569

Combine like terms: -7569 + 7569 = 0
0 + 232r + r2 = 0 + 7569
232r + r2 = 0 + 7569

Combine like terms: 0 + 7569 = 7569
232r + r2 = 7569

The r term is 232r.  Take half its coefficient (116).
Square it (13456) and add it to both sides.

Add '13456' to each side of the equation.
232r + 13456 + r2 = 7569 + 13456

Reorder the terms:
13456 + 232r + r2 = 7569 + 13456

Combine like terms: 7569 + 13456 = 21025
13456 + 232r + r2 = 21025

Factor a perfect square on the left side:
(r + 116)(r + 116) = 21025

Calculate the square root of the right side: 145

Break this problem into two subproblems by setting 
(r + 116) equal to 145 and -145.


Subproblem 1
r + 116 = 145

Simplifying
r + 116 = 145

Reorder the terms:
116 + r = 145

Solving
116 + r = 145

Solving for variable 'r'.

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

Add '-116' to each side of the equation.
116 + -116 + r = 145 + -116

Combine like terms: 116 + -116 = 0
0 + r = 145 + -116
r = 145 + -116

Combine like terms: 145 + -116 = 29
r = 29

Simplifying
r = 29


Subproblem 2
r + 116 = -145

Simplifying
r + 116 = -145

Reorder the terms:
116 + r = -145

Solving
116 + r = -145

Solving for variable 'r'.

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

Add '-116' to each side of the equation.
116 + -116 + r = -145 + -116

Combine like terms: 116 + -116 = 0
0 + r = -145 + -116
r = -145 + -116

Combine like terms: -145 + -116 = -261
r = -261

Simplifying
r = -261


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {29, -261}

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