502.4=12(3.141592654)(r)+(3.141592654)(r^2)

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Solution for 502.4=12(3.141592654)(r)+(3.141592654)(r^2) equation:


Simplifying
502.4 = 12(3.141592654)(r) + (3.141592654)(r2)

Multiply 12 * 3.141592654
502.4 = 37.69911185r + (3.141592654)(r2)

Solving
502.4 = 37.69911185r + 3.141592654r2

Solving for variable 'r'.

Reorder the terms:
502.4 + -37.69911185r + -3.141592654r2 = 37.69911185r + -37.69911185r + 3.141592654r2 + -3.141592654r2

Combine like terms: 37.69911185r + -37.69911185r = 0.00000000
502.4 + -37.69911185r + -3.141592654r2 = 0.00000000 + 3.141592654r2 + -3.141592654r2
502.4 + -37.69911185r + -3.141592654r2 = 3.141592654r2 + -3.141592654r2

Combine like terms: 3.141592654r2 + -3.141592654r2 = 0.000000000
502.4 + -37.69911185r + -3.141592654r2 = 0.000000000

Begin completing the square.  Divide all terms by
-3.141592654 the coefficient of the squared term: 

Divide each side by '-3.141592654'.
-159.9188868 + 12r + r2 = 0

Move the constant term to the right:

Add '159.9188868' to each side of the equation.
-159.9188868 + 12r + 159.9188868 + r2 = 0 + 159.9188868

Reorder the terms:
-159.9188868 + 159.9188868 + 12r + r2 = 0 + 159.9188868

Combine like terms: -159.9188868 + 159.9188868 = 0.0000000
0.0000000 + 12r + r2 = 0 + 159.9188868
12r + r2 = 0 + 159.9188868

Combine like terms: 0 + 159.9188868 = 159.9188868
12r + r2 = 159.9188868

The r term is 12r.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12r + 36 + r2 = 159.9188868 + 36

Reorder the terms:
36 + 12r + r2 = 159.9188868 + 36

Combine like terms: 159.9188868 + 36 = 195.9188868
36 + 12r + r2 = 195.9188868

Factor a perfect square on the left side:
(r + 6)(r + 6) = 195.9188868

Calculate the square root of the right side: 13.9971028

Break this problem into two subproblems by setting 
(r + 6) equal to 13.9971028 and -13.9971028.

Subproblem 1

r + 6 = 13.9971028 Simplifying r + 6 = 13.9971028 Reorder the terms: 6 + r = 13.9971028 Solving 6 + r = 13.9971028 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = 13.9971028 + -6 Combine like terms: 6 + -6 = 0 0 + r = 13.9971028 + -6 r = 13.9971028 + -6 Combine like terms: 13.9971028 + -6 = 7.9971028 r = 7.9971028 Simplifying r = 7.9971028

Subproblem 2

r + 6 = -13.9971028 Simplifying r + 6 = -13.9971028 Reorder the terms: 6 + r = -13.9971028 Solving 6 + r = -13.9971028 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = -13.9971028 + -6 Combine like terms: 6 + -6 = 0 0 + r = -13.9971028 + -6 r = -13.9971028 + -6 Combine like terms: -13.9971028 + -6 = -19.9971028 r = -19.9971028 Simplifying r = -19.9971028

Solution

The solution to the problem is based on the solutions from the subproblems. r = {7.9971028, -19.9971028}

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