406.8184=2*3.14*r^2+2*3.14*r*16

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Solution for 406.8184=2*3.14*r^2+2*3.14*r*16 equation:


Simplifying
406.8184 = 2 * 3.14r2 + 2 * 3.14r * 16

Multiply 2 * 3.14
406.8184 = 6.28r2 + 2 * 3.14r * 16

Reorder the terms for easier multiplication:
406.8184 = 6.28r2 + 2 * 3.14 * 16r

Multiply 2 * 3.14
406.8184 = 6.28r2 + 6.28 * 16r

Multiply 6.28 * 16
406.8184 = 6.28r2 + 100.48r

Reorder the terms:
406.8184 = 100.48r + 6.28r2

Solving
406.8184 = 100.48r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
406.8184 + -100.48r + -6.28r2 = 100.48r + -100.48r + 6.28r2 + -6.28r2

Combine like terms: 100.48r + -100.48r = 0.00
406.8184 + -100.48r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2
406.8184 + -100.48r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
406.8184 + -100.48r + -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'.
-64.78 + 16r + r2 = 0

Move the constant term to the right:

Add '64.78' to each side of the equation.
-64.78 + 16r + 64.78 + r2 = 0 + 64.78

Reorder the terms:
-64.78 + 64.78 + 16r + r2 = 0 + 64.78

Combine like terms: -64.78 + 64.78 = 0.00
0.00 + 16r + r2 = 0 + 64.78
16r + r2 = 0 + 64.78

Combine like terms: 0 + 64.78 = 64.78
16r + r2 = 64.78

The r term is 16r.  Take half its coefficient (8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
16r + 64 + r2 = 64.78 + 64

Reorder the terms:
64 + 16r + r2 = 64.78 + 64

Combine like terms: 64.78 + 64 = 128.78
64 + 16r + r2 = 128.78

Factor a perfect square on the left side:
(r + 8)(r + 8) = 128.78

Calculate the square root of the right side: 11.348127599

Break this problem into two subproblems by setting 
(r + 8) equal to 11.348127599 and -11.348127599.

Subproblem 1

r + 8 = 11.348127599 Simplifying r + 8 = 11.348127599 Reorder the terms: 8 + r = 11.348127599 Solving 8 + r = 11.348127599 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + r = 11.348127599 + -8 Combine like terms: 8 + -8 = 0 0 + r = 11.348127599 + -8 r = 11.348127599 + -8 Combine like terms: 11.348127599 + -8 = 3.348127599 r = 3.348127599 Simplifying r = 3.348127599

Subproblem 2

r + 8 = -11.348127599 Simplifying r + 8 = -11.348127599 Reorder the terms: 8 + r = -11.348127599 Solving 8 + r = -11.348127599 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + r = -11.348127599 + -8 Combine like terms: 8 + -8 = 0 0 + r = -11.348127599 + -8 r = -11.348127599 + -8 Combine like terms: -11.348127599 + -8 = -19.348127599 r = -19.348127599 Simplifying r = -19.348127599

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.348127599, -19.348127599}

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