9(a+3)a=6

Solution for 9(a+3)a=6 equation:

Simplifying
9(a + 3) * a = 6

Reorder the terms:
9(3 + a) * a = 6

Reorder the terms for easier multiplication:
9a(3 + a) = 6
(3 * 9a + a * 9a) = 6
(27a + 9a2) = 6

Solving
27a + 9a2 = 6

Solving for variable 'a'.

Reorder the terms:
-6 + 27a + 9a2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 27a + 9a2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-2 + 9a + 3a2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-2 + 9a + 3a2)' equal to zero and attempt to solve:

Simplifying
-2 + 9a + 3a2 = 0

Solving
-2 + 9a + 3a2 = 0

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

Divide each side by '3'.
-0.6666666667 + 3a + a2 = 0

Move the constant term to the right:

Add '0.6666666667' to each side of the equation.
-0.6666666667 + 3a + 0.6666666667 + a2 = 0 + 0.6666666667

Reorder the terms:
-0.6666666667 + 0.6666666667 + 3a + a2 = 0 + 0.6666666667

Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000
0.0000000000 + 3a + a2 = 0 + 0.6666666667
3a + a2 = 0 + 0.6666666667

Combine like terms: 0 + 0.6666666667 = 0.6666666667
3a + a2 = 0.6666666667

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = 0.6666666667 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 0.6666666667 + 2.25

Combine like terms: 0.6666666667 + 2.25 = 2.9166666667
2.25 + 3a + a2 = 2.9166666667

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 2.9166666667

Calculate the square root of the right side: 1.707825128

Break this problem into two subproblems by setting 
(a + 1.5) equal to 1.707825128 and -1.707825128.

Subproblem 1
a + 1.5 = 1.707825128

Simplifying
a + 1.5 = 1.707825128

Reorder the terms:
1.5 + a = 1.707825128

Solving
1.5 + a = 1.707825128

Solving for variable 'a'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + a = 1.707825128 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = 1.707825128 + -1.5
a = 1.707825128 + -1.5

Combine like terms: 1.707825128 + -1.5 = 0.207825128
a = 0.207825128

Simplifying
a = 0.207825128

Subproblem 2
a + 1.5 = -1.707825128

Simplifying
a + 1.5 = -1.707825128

Reorder the terms:
1.5 + a = -1.707825128

Solving
1.5 + a = -1.707825128

Solving for variable 'a'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + a = -1.707825128 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = -1.707825128 + -1.5
a = -1.707825128 + -1.5

Combine like terms: -1.707825128 + -1.5 = -3.207825128
a = -3.207825128

Simplifying
a = -3.207825128

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.207825128, -3.207825128}
Solution
a = {0.207825128, -3.207825128}