2*3.14*r^2+2*3.14*r*5.3=285.9918

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Solution for 2*3.14*r^2+2*3.14*r*5.3=285.9918 equation:


Simplifying
2 * 3.14r2 + 2 * 3.14r * 5.3 = 285.9918

Multiply 2 * 3.14
6.28r2 + 2 * 3.14r * 5.3 = 285.9918

Reorder the terms for easier multiplication:
6.28r2 + 2 * 3.14 * 5.3r = 285.9918

Multiply 2 * 3.14
6.28r2 + 6.28 * 5.3r = 285.9918

Multiply 6.28 * 5.3
6.28r2 + 33.284r = 285.9918

Reorder the terms:
33.284r + 6.28r2 = 285.9918

Solving
33.284r + 6.28r2 = 285.9918

Solving for variable 'r'.

Reorder the terms:
-285.9918 + 33.284r + 6.28r2 = 285.9918 + -285.9918

Combine like terms: 285.9918 + -285.9918 = 0.0000
-285.9918 + 33.284r + 6.28r2 = 0.0000

Begin completing the square.  Divide all terms by
6.28 the coefficient of the squared term: 

Divide each side by '6.28'.
-45.54009554 + 5.3r + r2 = 0.00

Move the constant term to the right:

Add '45.54009554' to each side of the equation.
-45.54009554 + 5.3r + 45.54009554 + r2 = 0.00 + 45.54009554

Reorder the terms:
-45.54009554 + 45.54009554 + 5.3r + r2 = 0.00 + 45.54009554

Combine like terms: -45.54009554 + 45.54009554 = 0.00000000
0.00000000 + 5.3r + r2 = 0.00 + 45.54009554
5.3r + r2 = 0.00 + 45.54009554

Combine like terms: 0.00 + 45.54009554 = 45.54009554
5.3r + r2 = 45.54009554

The r term is 5.3r.  Take half its coefficient (2.65).
Square it (7.0225) and add it to both sides.

Add '7.0225' to each side of the equation.
5.3r + 7.0225 + r2 = 45.54009554 + 7.0225

Reorder the terms:
7.0225 + 5.3r + r2 = 45.54009554 + 7.0225

Combine like terms: 45.54009554 + 7.0225 = 52.56259554
7.0225 + 5.3r + r2 = 52.56259554

Factor a perfect square on the left side:
(r + 2.65)(r + 2.65) = 52.56259554

Calculate the square root of the right side: 7.250006589

Break this problem into two subproblems by setting 
(r + 2.65) equal to 7.250006589 and -7.250006589.

Subproblem 1

r + 2.65 = 7.250006589 Simplifying r + 2.65 = 7.250006589 Reorder the terms: 2.65 + r = 7.250006589 Solving 2.65 + r = 7.250006589 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = 7.250006589 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = 7.250006589 + -2.65 r = 7.250006589 + -2.65 Combine like terms: 7.250006589 + -2.65 = 4.600006589 r = 4.600006589 Simplifying r = 4.600006589

Subproblem 2

r + 2.65 = -7.250006589 Simplifying r + 2.65 = -7.250006589 Reorder the terms: 2.65 + r = -7.250006589 Solving 2.65 + r = -7.250006589 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = -7.250006589 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = -7.250006589 + -2.65 r = -7.250006589 + -2.65 Combine like terms: -7.250006589 + -2.65 = -9.900006589 r = -9.900006589 Simplifying r = -9.900006589

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.600006589, -9.900006589}

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