1909=2*3.14*r^2+2*3.14*r*30

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Solution for 1909=2*3.14*r^2+2*3.14*r*30 equation:


Simplifying
1909 = 2 * 3.14r2 + 2 * 3.14r * 30

Multiply 2 * 3.14
1909 = 6.28r2 + 2 * 3.14r * 30

Reorder the terms for easier multiplication:
1909 = 6.28r2 + 2 * 3.14 * 30r

Multiply 2 * 3.14
1909 = 6.28r2 + 6.28 * 30r

Multiply 6.28 * 30
1909 = 6.28r2 + 188.4r

Reorder the terms:
1909 = 188.4r + 6.28r2

Solving
1909 = 188.4r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
1909 + -188.4r + -6.28r2 = 188.4r + -188.4r + 6.28r2 + -6.28r2

Combine like terms: 188.4r + -188.4r = 0.0
1909 + -188.4r + -6.28r2 = 0.0 + 6.28r2 + -6.28r2
1909 + -188.4r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
1909 + -188.4r + -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'.
-303.9808917 + 30r + r2 = 0

Move the constant term to the right:

Add '303.9808917' to each side of the equation.
-303.9808917 + 30r + 303.9808917 + r2 = 0 + 303.9808917

Reorder the terms:
-303.9808917 + 303.9808917 + 30r + r2 = 0 + 303.9808917

Combine like terms: -303.9808917 + 303.9808917 = 0.0000000
0.0000000 + 30r + r2 = 0 + 303.9808917
30r + r2 = 0 + 303.9808917

Combine like terms: 0 + 303.9808917 = 303.9808917
30r + r2 = 303.9808917

The r term is 30r.  Take half its coefficient (15).
Square it (225) and add it to both sides.

Add '225' to each side of the equation.
30r + 225 + r2 = 303.9808917 + 225

Reorder the terms:
225 + 30r + r2 = 303.9808917 + 225

Combine like terms: 303.9808917 + 225 = 528.9808917
225 + 30r + r2 = 528.9808917

Factor a perfect square on the left side:
(r + 15)(r + 15) = 528.9808917

Calculate the square root of the right side: 22.999584598

Break this problem into two subproblems by setting 
(r + 15) equal to 22.999584598 and -22.999584598.

Subproblem 1

r + 15 = 22.999584598 Simplifying r + 15 = 22.999584598 Reorder the terms: 15 + r = 22.999584598 Solving 15 + r = 22.999584598 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + r = 22.999584598 + -15 Combine like terms: 15 + -15 = 0 0 + r = 22.999584598 + -15 r = 22.999584598 + -15 Combine like terms: 22.999584598 + -15 = 7.999584598 r = 7.999584598 Simplifying r = 7.999584598

Subproblem 2

r + 15 = -22.999584598 Simplifying r + 15 = -22.999584598 Reorder the terms: 15 + r = -22.999584598 Solving 15 + r = -22.999584598 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + r = -22.999584598 + -15 Combine like terms: 15 + -15 = 0 0 + r = -22.999584598 + -15 r = -22.999584598 + -15 Combine like terms: -22.999584598 + -15 = -37.999584598 r = -37.999584598 Simplifying r = -37.999584598

Solution

The solution to the problem is based on the solutions from the subproblems. r = {7.999584598, -37.999584598}

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