8(36+r)r=12

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Solution for 8(36+r)r=12 equation: 


Simplifying
8(36 + r) * r = 12

Reorder the terms for easier multiplication:
8r(36 + r) = 12
(36 * 8r + r * 8r) = 12
(288r + 8r2) = 12

Solving
288r + 8r2 = 12

Solving for variable 'r'.

Reorder the terms:
-12 + 288r + 8r2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 288r + 8r2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-3 + 72r + 2r2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-3 + 72r + 2r2)' equal to zero and attempt to solve:

Simplifying
-3 + 72r + 2r2 = 0

Solving
-3 + 72r + 2r2 = 0

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

Divide each side by '2'.
-1.5 + 36r + r2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 36r + 1.5 + r2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 36r + r2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 36r + r2 = 0 + 1.5
36r + r2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
36r + r2 = 1.5

The r term is 36r.  Take half its coefficient (18).
Square it (324) and add it to both sides.

Add '324' to each side of the equation.
36r + 324 + r2 = 1.5 + 324

Reorder the terms:
324 + 36r + r2 = 1.5 + 324

Combine like terms: 1.5 + 324 = 325.5
324 + 36r + r2 = 325.5

Factor a perfect square on the left side:
(r + 18)(r + 18) = 325.5

Calculate the square root of the right side: 18.041618553

Break this problem into two subproblems by setting 
(r + 18) equal to 18.041618553 and -18.041618553.


Subproblem 1
r + 18 = 18.041618553

Simplifying
r + 18 = 18.041618553

Reorder the terms:
18 + r = 18.041618553

Solving
18 + r = 18.041618553

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-18' to each side of the equation.
18 + -18 + r = 18.041618553 + -18

Combine like terms: 18 + -18 = 0
0 + r = 18.041618553 + -18
r = 18.041618553 + -18

Combine like terms: 18.041618553 + -18 = 0.041618553
r = 0.041618553

Simplifying
r = 0.041618553


Subproblem 2
r + 18 = -18.041618553

Simplifying
r + 18 = -18.041618553

Reorder the terms:
18 + r = -18.041618553

Solving
18 + r = -18.041618553

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-18' to each side of the equation.
18 + -18 + r = -18.041618553 + -18

Combine like terms: 18 + -18 = 0
0 + r = -18.041618553 + -18
r = -18.041618553 + -18

Combine like terms: -18.041618553 + -18 = -36.041618553
r = -36.041618553

Simplifying
r = -36.041618553


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {0.041618553, -36.041618553}
Solution
r = {0.041618553, -36.041618553}

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