414.48=2*3.14*r^2+2*3.14*r*5.5

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Solution for 414.48=2*3.14*r^2+2*3.14*r*5.5 equation:


Simplifying
414.48 = 2 * 3.14r2 + 2 * 3.14r * 5.5

Multiply 2 * 3.14
414.48 = 6.28r2 + 2 * 3.14r * 5.5

Reorder the terms for easier multiplication:
414.48 = 6.28r2 + 2 * 3.14 * 5.5r

Multiply 2 * 3.14
414.48 = 6.28r2 + 6.28 * 5.5r

Multiply 6.28 * 5.5
414.48 = 6.28r2 + 34.54r

Reorder the terms:
414.48 = 34.54r + 6.28r2

Solving
414.48 = 34.54r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
414.48 + -34.54r + -6.28r2 = 34.54r + -34.54r + 6.28r2 + -6.28r2

Combine like terms: 34.54r + -34.54r = 0.00
414.48 + -34.54r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2
414.48 + -34.54r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
414.48 + -34.54r + -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'.
-66 + 5.5r + r2 = 0

Move the constant term to the right:

Add '66' to each side of the equation.
-66 + 5.5r + 66 + r2 = 0 + 66

Reorder the terms:
-66 + 66 + 5.5r + r2 = 0 + 66

Combine like terms: -66 + 66 = 0
0 + 5.5r + r2 = 0 + 66
5.5r + r2 = 0 + 66

Combine like terms: 0 + 66 = 66
5.5r + r2 = 66

The r term is 5.5r.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5r + 7.5625 + r2 = 66 + 7.5625

Reorder the terms:
7.5625 + 5.5r + r2 = 66 + 7.5625

Combine like terms: 66 + 7.5625 = 73.5625
7.5625 + 5.5r + r2 = 73.5625

Factor a perfect square on the left side:
(r + 2.75)(r + 2.75) = 73.5625

Calculate the square root of the right side: 8.576858399

Break this problem into two subproblems by setting 
(r + 2.75) equal to 8.576858399 and -8.576858399.

Subproblem 1

r + 2.75 = 8.576858399 Simplifying r + 2.75 = 8.576858399 Reorder the terms: 2.75 + r = 8.576858399 Solving 2.75 + r = 8.576858399 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + r = 8.576858399 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + r = 8.576858399 + -2.75 r = 8.576858399 + -2.75 Combine like terms: 8.576858399 + -2.75 = 5.826858399 r = 5.826858399 Simplifying r = 5.826858399

Subproblem 2

r + 2.75 = -8.576858399 Simplifying r + 2.75 = -8.576858399 Reorder the terms: 2.75 + r = -8.576858399 Solving 2.75 + r = -8.576858399 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + r = -8.576858399 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + r = -8.576858399 + -2.75 r = -8.576858399 + -2.75 Combine like terms: -8.576858399 + -2.75 = -11.326858399 r = -11.326858399 Simplifying r = -11.326858399

Solution

The solution to the problem is based on the solutions from the subproblems. r = {5.826858399, -11.326858399}

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