512.3854=2*3.14*r^2+2*3.14*r*15.8

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Solution for 512.3854=2*3.14*r^2+2*3.14*r*15.8 equation:


Simplifying
512.3854 = 2 * 3.14r2 + 2 * 3.14r * 15.8

Multiply 2 * 3.14
512.3854 = 6.28r2 + 2 * 3.14r * 15.8

Reorder the terms for easier multiplication:
512.3854 = 6.28r2 + 2 * 3.14 * 15.8r

Multiply 2 * 3.14
512.3854 = 6.28r2 + 6.28 * 15.8r

Multiply 6.28 * 15.8
512.3854 = 6.28r2 + 99.224r

Reorder the terms:
512.3854 = 99.224r + 6.28r2

Solving
512.3854 = 99.224r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
512.3854 + -99.224r + -6.28r2 = 99.224r + -99.224r + 6.28r2 + -6.28r2

Combine like terms: 99.224r + -99.224r = 0.000
512.3854 + -99.224r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
512.3854 + -99.224r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
512.3854 + -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'.
-81.59003185 + 15.8r + r2 = 0

Move the constant term to the right:

Add '81.59003185' to each side of the equation.
-81.59003185 + 15.8r + 81.59003185 + r2 = 0 + 81.59003185

Reorder the terms:
-81.59003185 + 81.59003185 + 15.8r + r2 = 0 + 81.59003185

Combine like terms: -81.59003185 + 81.59003185 = 0.00000000
0.00000000 + 15.8r + r2 = 0 + 81.59003185
15.8r + r2 = 0 + 81.59003185

Combine like terms: 0 + 81.59003185 = 81.59003185
15.8r + r2 = 81.59003185

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 = 81.59003185 + 62.41

Reorder the terms:
62.41 + 15.8r + r2 = 81.59003185 + 62.41

Combine like terms: 81.59003185 + 62.41 = 144.00003185
62.41 + 15.8r + r2 = 144.00003185

Factor a perfect square on the left side:
(r + 7.9)(r + 7.9) = 144.00003185

Calculate the square root of the right side: 12.000001327

Break this problem into two subproblems by setting 
(r + 7.9) equal to 12.000001327 and -12.000001327.

Subproblem 1

r + 7.9 = 12.000001327 Simplifying r + 7.9 = 12.000001327 Reorder the terms: 7.9 + r = 12.000001327 Solving 7.9 + r = 12.000001327 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 = 12.000001327 + -7.9 Combine like terms: 7.9 + -7.9 = 0.0 0.0 + r = 12.000001327 + -7.9 r = 12.000001327 + -7.9 Combine like terms: 12.000001327 + -7.9 = 4.100001327 r = 4.100001327 Simplifying r = 4.100001327

Subproblem 2

r + 7.9 = -12.000001327 Simplifying r + 7.9 = -12.000001327 Reorder the terms: 7.9 + r = -12.000001327 Solving 7.9 + r = -12.000001327 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 = -12.000001327 + -7.9 Combine like terms: 7.9 + -7.9 = 0.0 0.0 + r = -12.000001327 + -7.9 r = -12.000001327 + -7.9 Combine like terms: -12.000001327 + -7.9 = -19.900001327 r = -19.900001327 Simplifying r = -19.900001327

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.100001327, -19.900001327}

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