0=6.28r^2+113.668r-621.72

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Solution for 0=6.28r^2+113.668r-621.72 equation:


Simplifying
0 = 6.28r2 + 113.668r + -621.72

Reorder the terms:
0 = -621.72 + 113.668r + 6.28r2

Solving
0 = -621.72 + 113.668r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 621.72 = 621.72
621.72 + -113.668r + -6.28r2 = -621.72 + 113.668r + 6.28r2 + 621.72 + -113.668r + -6.28r2

Reorder the terms:
621.72 + -113.668r + -6.28r2 = -621.72 + 621.72 + 113.668r + -113.668r + 6.28r2 + -6.28r2

Combine like terms: -621.72 + 621.72 = 0.00
621.72 + -113.668r + -6.28r2 = 0.00 + 113.668r + -113.668r + 6.28r2 + -6.28r2
621.72 + -113.668r + -6.28r2 = 113.668r + -113.668r + 6.28r2 + -6.28r2

Combine like terms: 113.668r + -113.668r = 0.000
621.72 + -113.668r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
621.72 + -113.668r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
621.72 + -113.668r + -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'.
-99 + 18.1r + r2 = 0

Move the constant term to the right:

Add '99' to each side of the equation.
-99 + 18.1r + 99 + r2 = 0 + 99

Reorder the terms:
-99 + 99 + 18.1r + r2 = 0 + 99

Combine like terms: -99 + 99 = 0
0 + 18.1r + r2 = 0 + 99
18.1r + r2 = 0 + 99

Combine like terms: 0 + 99 = 99
18.1r + r2 = 99

The r term is 18.1r.  Take half its coefficient (9.05).
Square it (81.9025) and add it to both sides.

Add '81.9025' to each side of the equation.
18.1r + 81.9025 + r2 = 99 + 81.9025

Reorder the terms:
81.9025 + 18.1r + r2 = 99 + 81.9025

Combine like terms: 99 + 81.9025 = 180.9025
81.9025 + 18.1r + r2 = 180.9025

Factor a perfect square on the left side:
(r + 9.05)(r + 9.05) = 180.9025

Calculate the square root of the right side: 13.45

Break this problem into two subproblems by setting 
(r + 9.05) equal to 13.45 and -13.45.

Subproblem 1

r + 9.05 = 13.45 Simplifying r + 9.05 = 13.45 Reorder the terms: 9.05 + r = 13.45 Solving 9.05 + r = 13.45 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-9.05' to each side of the equation. 9.05 + -9.05 + r = 13.45 + -9.05 Combine like terms: 9.05 + -9.05 = 0.00 0.00 + r = 13.45 + -9.05 r = 13.45 + -9.05 Combine like terms: 13.45 + -9.05 = 4.4 r = 4.4 Simplifying r = 4.4

Subproblem 2

r + 9.05 = -13.45 Simplifying r + 9.05 = -13.45 Reorder the terms: 9.05 + r = -13.45 Solving 9.05 + r = -13.45 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-9.05' to each side of the equation. 9.05 + -9.05 + r = -13.45 + -9.05 Combine like terms: 9.05 + -9.05 = 0.00 0.00 + r = -13.45 + -9.05 r = -13.45 + -9.05 Combine like terms: -13.45 + -9.05 = -22.5 r = -22.5 Simplifying r = -22.5

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.4, -22.5}

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