0=6.28r^2+1224.6r-16981.12

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Solution for 0=6.28r^2+1224.6r-16981.12 equation:


Simplifying
0 = 6.28r2 + 1224.6r + -16981.12

Reorder the terms:
0 = -16981.12 + 1224.6r + 6.28r2

Solving
0 = -16981.12 + 1224.6r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 16981.12 = 16981.12
16981.12 + -1224.6r + -6.28r2 = -16981.12 + 1224.6r + 6.28r2 + 16981.12 + -1224.6r + -6.28r2

Reorder the terms:
16981.12 + -1224.6r + -6.28r2 = -16981.12 + 16981.12 + 1224.6r + -1224.6r + 6.28r2 + -6.28r2

Combine like terms: -16981.12 + 16981.12 = 0.00
16981.12 + -1224.6r + -6.28r2 = 0.00 + 1224.6r + -1224.6r + 6.28r2 + -6.28r2
16981.12 + -1224.6r + -6.28r2 = 1224.6r + -1224.6r + 6.28r2 + -6.28r2

Combine like terms: 1224.6r + -1224.6r = 0.0
16981.12 + -1224.6r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2
16981.12 + -1224.6r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
16981.12 + -1224.6r + -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'.
-2704 + 195r + r2 = 0

Move the constant term to the right:

Add '2704' to each side of the equation.
-2704 + 195r + 2704 + r2 = 0 + 2704

Reorder the terms:
-2704 + 2704 + 195r + r2 = 0 + 2704

Combine like terms: -2704 + 2704 = 0
0 + 195r + r2 = 0 + 2704
195r + r2 = 0 + 2704

Combine like terms: 0 + 2704 = 2704
195r + r2 = 2704

The r term is 195r.  Take half its coefficient (97.5).
Square it (9506.25) and add it to both sides.

Add '9506.25' to each side of the equation.
195r + 9506.25 + r2 = 2704 + 9506.25

Reorder the terms:
9506.25 + 195r + r2 = 2704 + 9506.25

Combine like terms: 2704 + 9506.25 = 12210.25
9506.25 + 195r + r2 = 12210.25

Factor a perfect square on the left side:
(r + 97.5)(r + 97.5) = 12210.25

Calculate the square root of the right side: 110.5

Break this problem into two subproblems by setting 
(r + 97.5) equal to 110.5 and -110.5.

Subproblem 1

r + 97.5 = 110.5 Simplifying r + 97.5 = 110.5 Reorder the terms: 97.5 + r = 110.5 Solving 97.5 + r = 110.5 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-97.5' to each side of the equation. 97.5 + -97.5 + r = 110.5 + -97.5 Combine like terms: 97.5 + -97.5 = 0.0 0.0 + r = 110.5 + -97.5 r = 110.5 + -97.5 Combine like terms: 110.5 + -97.5 = 13 r = 13 Simplifying r = 13

Subproblem 2

r + 97.5 = -110.5 Simplifying r + 97.5 = -110.5 Reorder the terms: 97.5 + r = -110.5 Solving 97.5 + r = -110.5 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-97.5' to each side of the equation. 97.5 + -97.5 + r = -110.5 + -97.5 Combine like terms: 97.5 + -97.5 = 0.0 0.0 + r = -110.5 + -97.5 r = -110.5 + -97.5 Combine like terms: -110.5 + -97.5 = -208 r = -208 Simplifying r = -208

Solution

The solution to the problem is based on the solutions from the subproblems. r = {13, -208}

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