512.3852=2*3.14*r^2+2*3.14*r*16.2

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Solution for 512.3852=2*3.14*r^2+2*3.14*r*16.2 equation:


Simplifying
512.3852 = 2 * 3.14r2 + 2 * 3.14r * 16.2

Multiply 2 * 3.14
512.3852 = 6.28r2 + 2 * 3.14r * 16.2

Reorder the terms for easier multiplication:
512.3852 = 6.28r2 + 2 * 3.14 * 16.2r

Multiply 2 * 3.14
512.3852 = 6.28r2 + 6.28 * 16.2r

Multiply 6.28 * 16.2
512.3852 = 6.28r2 + 101.736r

Reorder the terms:
512.3852 = 101.736r + 6.28r2

Solving
512.3852 = 101.736r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
512.3852 + -101.736r + -6.28r2 = 101.736r + -101.736r + 6.28r2 + -6.28r2

Combine like terms: 101.736r + -101.736r = 0.000
512.3852 + -101.736r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
512.3852 + -101.736r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
512.3852 + -101.736r + -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 + 16.2r + r2 = 0

Move the constant term to the right:

Add '81.59' to each side of the equation.
-81.59 + 16.2r + 81.59 + r2 = 0 + 81.59

Reorder the terms:
-81.59 + 81.59 + 16.2r + r2 = 0 + 81.59

Combine like terms: -81.59 + 81.59 = 0.00
0.00 + 16.2r + r2 = 0 + 81.59
16.2r + r2 = 0 + 81.59

Combine like terms: 0 + 81.59 = 81.59
16.2r + r2 = 81.59

The r term is 16.2r.  Take half its coefficient (8.1).
Square it (65.61) and add it to both sides.

Add '65.61' to each side of the equation.
16.2r + 65.61 + r2 = 81.59 + 65.61

Reorder the terms:
65.61 + 16.2r + r2 = 81.59 + 65.61

Combine like terms: 81.59 + 65.61 = 147.2
65.61 + 16.2r + r2 = 147.2

Factor a perfect square on the left side:
(r + 8.1)(r + 8.1) = 147.2

Calculate the square root of the right side: 12.13260071

Break this problem into two subproblems by setting 
(r + 8.1) equal to 12.13260071 and -12.13260071.

Subproblem 1

r + 8.1 = 12.13260071 Simplifying r + 8.1 = 12.13260071 Reorder the terms: 8.1 + r = 12.13260071 Solving 8.1 + r = 12.13260071 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.1' to each side of the equation. 8.1 + -8.1 + r = 12.13260071 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + r = 12.13260071 + -8.1 r = 12.13260071 + -8.1 Combine like terms: 12.13260071 + -8.1 = 4.03260071 r = 4.03260071 Simplifying r = 4.03260071

Subproblem 2

r + 8.1 = -12.13260071 Simplifying r + 8.1 = -12.13260071 Reorder the terms: 8.1 + r = -12.13260071 Solving 8.1 + r = -12.13260071 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-8.1' to each side of the equation. 8.1 + -8.1 + r = -12.13260071 + -8.1 Combine like terms: 8.1 + -8.1 = 0.0 0.0 + r = -12.13260071 + -8.1 r = -12.13260071 + -8.1 Combine like terms: -12.13260071 + -8.1 = -20.23260071 r = -20.23260071 Simplifying r = -20.23260071

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.03260071, -20.23260071}

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