512.3852=2*3.14*r^2+2*3.14*r*15.8

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Solution for 512.3852=2*3.14*r^2+2*3.14*r*15.8 equation: 


Simplifying
512.3852 = 2 * 3.14r2 + 2 * 3.14r * 15.8

Multiply 2 * 3.14
512.3852 = 6.28r2 + 2 * 3.14r * 15.8

Reorder the terms for easier multiplication:
512.3852 = 6.28r2 + 2 * 3.14 * 15.8r

Multiply 2 * 3.14
512.3852 = 6.28r2 + 6.28 * 15.8r

Multiply 6.28 * 15.8
512.3852 = 6.28r2 + 99.224r

Reorder the terms:
512.3852 = 99.224r + 6.28r2

Solving
512.3852 = 99.224r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
512.3852 + -99.224r + -6.28r2 = 99.224r + -99.224r + 6.28r2 + -6.28r2

Combine like terms: 99.224r + -99.224r = 0.000
512.3852 + -99.224r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
512.3852 + -99.224r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
512.3852 + -99.224r + -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'.
-81.59 + 15.8r + r2 = 0

Move the constant term to the right:

Add '81.59' to each side of the equation.
-81.59 + 15.8r + 81.59 + r2 = 0 + 81.59

Reorder the terms:
-81.59 + 81.59 + 15.8r + r2 = 0 + 81.59

Combine like terms: -81.59 + 81.59 = 0.00
0.00 + 15.8r + r2 = 0 + 81.59
15.8r + r2 = 0 + 81.59

Combine like terms: 0 + 81.59 = 81.59
15.8r + r2 = 81.59

The r term is 15.8r.  Take half its coefficient (7.9).
Square it (62.41) and add it to both sides.

Add '62.41' to each side of the equation.
15.8r + 62.41 + r2 = 81.59 + 62.41

Reorder the terms:
62.41 + 15.8r + r2 = 81.59 + 62.41

Combine like terms: 81.59 + 62.41 = 144
62.41 + 15.8r + r2 = 144

Factor a perfect square on the left side:
(r + 7.9)(r + 7.9) = 144

Calculate the square root of the right side: 12

Break this problem into two subproblems by setting 
(r + 7.9) equal to 12 and -12.


Subproblem 1
r + 7.9 = 12

Simplifying
r + 7.9 = 12

Reorder the terms:
7.9 + r = 12

Solving
7.9 + r = 12

Solving for variable 'r'.

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

Add '-7.9' to each side of the equation.
7.9 + -7.9 + r = 12 + -7.9

Combine like terms: 7.9 + -7.9 = 0.0
0.0 + r = 12 + -7.9
r = 12 + -7.9

Combine like terms: 12 + -7.9 = 4.1
r = 4.1

Simplifying
r = 4.1


Subproblem 2
r + 7.9 = -12

Simplifying
r + 7.9 = -12

Reorder the terms:
7.9 + r = -12

Solving
7.9 + r = -12

Solving for variable 'r'.

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

Add '-7.9' to each side of the equation.
7.9 + -7.9 + r = -12 + -7.9

Combine like terms: 7.9 + -7.9 = 0.0
0.0 + r = -12 + -7.9
r = -12 + -7.9

Combine like terms: -12 + -7.9 = -19.9
r = -19.9

Simplifying
r = -19.9


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.1, -19.9}

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