509.8104=2*3.14*r^2+2*3.14*r*15.7

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Solution for 509.8104=2*3.14*r^2+2*3.14*r*15.7 equation: 


Simplifying
509.8104 = 2 * 3.14r2 + 2 * 3.14r * 15.7

Multiply 2 * 3.14
509.8104 = 6.28r2 + 2 * 3.14r * 15.7

Reorder the terms for easier multiplication:
509.8104 = 6.28r2 + 2 * 3.14 * 15.7r

Multiply 2 * 3.14
509.8104 = 6.28r2 + 6.28 * 15.7r

Multiply 6.28 * 15.7
509.8104 = 6.28r2 + 98.596r

Reorder the terms:
509.8104 = 98.596r + 6.28r2

Solving
509.8104 = 98.596r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
509.8104 + -98.596r + -6.28r2 = 98.596r + -98.596r + 6.28r2 + -6.28r2

Combine like terms: 98.596r + -98.596r = 0.000
509.8104 + -98.596r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
509.8104 + -98.596r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
509.8104 + -98.596r + -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'.
-81.18 + 15.7r + r2 = 0

Move the constant term to the right:

Add '81.18' to each side of the equation.
-81.18 + 15.7r + 81.18 + r2 = 0 + 81.18

Reorder the terms:
-81.18 + 81.18 + 15.7r + r2 = 0 + 81.18

Combine like terms: -81.18 + 81.18 = 0.00
0.00 + 15.7r + r2 = 0 + 81.18
15.7r + r2 = 0 + 81.18

Combine like terms: 0 + 81.18 = 81.18
15.7r + r2 = 81.18

The r term is 15.7r.  Take half its coefficient (7.85).
Square it (61.6225) and add it to both sides.

Add '61.6225' to each side of the equation.
15.7r + 61.6225 + r2 = 81.18 + 61.6225

Reorder the terms:
61.6225 + 15.7r + r2 = 81.18 + 61.6225

Combine like terms: 81.18 + 61.6225 = 142.8025
61.6225 + 15.7r + r2 = 142.8025

Factor a perfect square on the left side:
(r + 7.85)(r + 7.85) = 142.8025

Calculate the square root of the right side: 11.95

Break this problem into two subproblems by setting 
(r + 7.85) equal to 11.95 and -11.95.


Subproblem 1
r + 7.85 = 11.95

Simplifying
r + 7.85 = 11.95

Reorder the terms:
7.85 + r = 11.95

Solving
7.85 + r = 11.95

Solving for variable 'r'.

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

Add '-7.85' to each side of the equation.
7.85 + -7.85 + r = 11.95 + -7.85

Combine like terms: 7.85 + -7.85 = 0.00
0.00 + r = 11.95 + -7.85
r = 11.95 + -7.85

Combine like terms: 11.95 + -7.85 = 4.1
r = 4.1

Simplifying
r = 4.1


Subproblem 2
r + 7.85 = -11.95

Simplifying
r + 7.85 = -11.95

Reorder the terms:
7.85 + r = -11.95

Solving
7.85 + r = -11.95

Solving for variable 'r'.

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

Add '-7.85' to each side of the equation.
7.85 + -7.85 + r = -11.95 + -7.85

Combine like terms: 7.85 + -7.85 = 0.00
0.00 + r = -11.95 + -7.85
r = -11.95 + -7.85

Combine like terms: -11.95 + -7.85 = -19.8
r = -19.8

Simplifying
r = -19.8


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.1, -19.8}

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