2(9r^2+72r+67)=0

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Solution for 2(9r^2+72r+67)=0 equation: 


Simplifying
2(9r2 + 72r + 67) = 0

Reorder the terms:
2(67 + 72r + 9r2) = 0
(67 * 2 + 72r * 2 + 9r2 * 2) = 0
(134 + 144r + 18r2) = 0

Solving
134 + 144r + 18r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '2'.
2(67 + 72r + 9r2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(67 + 72r + 9r2)' equal to zero and attempt to solve:

Simplifying
67 + 72r + 9r2 = 0

Solving
67 + 72r + 9r2 = 0

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

Divide each side by '9'.
7.444444444 + 8r + r2 = 0

Move the constant term to the right:

Add '-7.444444444' to each side of the equation.
7.444444444 + 8r + -7.444444444 + r2 = 0 + -7.444444444

Reorder the terms:
7.444444444 + -7.444444444 + 8r + r2 = 0 + -7.444444444

Combine like terms: 7.444444444 + -7.444444444 = 0.000000000
0.000000000 + 8r + r2 = 0 + -7.444444444
8r + r2 = 0 + -7.444444444

Combine like terms: 0 + -7.444444444 = -7.444444444
8r + r2 = -7.444444444

The r term is 8r.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8r + 16 + r2 = -7.444444444 + 16

Reorder the terms:
16 + 8r + r2 = -7.444444444 + 16

Combine like terms: -7.444444444 + 16 = 8.555555556
16 + 8r + r2 = 8.555555556

Factor a perfect square on the left side:
(r + 4)(r + 4) = 8.555555556

Calculate the square root of the right side: 2.924988129

Break this problem into two subproblems by setting 
(r + 4) equal to 2.924988129 and -2.924988129.


Subproblem 1
r + 4 = 2.924988129

Simplifying
r + 4 = 2.924988129

Reorder the terms:
4 + r = 2.924988129

Solving
4 + r = 2.924988129

Solving for variable 'r'.

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

Add '-4' to each side of the equation.
4 + -4 + r = 2.924988129 + -4

Combine like terms: 4 + -4 = 0
0 + r = 2.924988129 + -4
r = 2.924988129 + -4

Combine like terms: 2.924988129 + -4 = -1.075011871
r = -1.075011871

Simplifying
r = -1.075011871


Subproblem 2
r + 4 = -2.924988129

Simplifying
r + 4 = -2.924988129

Reorder the terms:
4 + r = -2.924988129

Solving
4 + r = -2.924988129

Solving for variable 'r'.

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

Add '-4' to each side of the equation.
4 + -4 + r = -2.924988129 + -4

Combine like terms: 4 + -4 = 0
0 + r = -2.924988129 + -4
r = -2.924988129 + -4

Combine like terms: -2.924988129 + -4 = -6.924988129
r = -6.924988129

Simplifying
r = -6.924988129


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-1.075011871, -6.924988129}
Solution
r = {-1.075011871, -6.924988129}

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