5x(6x+3)=9

Solution for 5x(6x+3)=9 equation:

Simplifying
5x(6x + 3) = 9

Reorder the terms:
5x(3 + 6x) = 9
(3 * 5x + 6x * 5x) = 9
(15x + 30x2) = 9

Solving
15x + 30x2 = 9

Solving for variable 'x'.

Reorder the terms:
-9 + 15x + 30x2 = 9 + -9

Combine like terms: 9 + -9 = 0
-9 + 15x + 30x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + 5x + 10x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-3 + 5x + 10x2)' equal to zero and attempt to solve:

Simplifying
-3 + 5x + 10x2 = 0

Solving
-3 + 5x + 10x2 = 0

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

Divide each side by '10'.
-0.3 + 0.5x + x2 = 0.0

Move the constant term to the right:

Add '0.3' to each side of the equation.
-0.3 + 0.5x + 0.3 + x2 = 0.0 + 0.3

Reorder the terms:
-0.3 + 0.3 + 0.5x + x2 = 0.0 + 0.3

Combine like terms: -0.3 + 0.3 = 0.0
0.0 + 0.5x + x2 = 0.0 + 0.3
0.5x + x2 = 0.0 + 0.3

Combine like terms: 0.0 + 0.3 = 0.3
0.5x + x2 = 0.3

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 0.3 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 0.3 + 0.0625

Combine like terms: 0.3 + 0.0625 = 0.3625
0.0625 + 0.5x + x2 = 0.3625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.3625

Calculate the square root of the right side: 0.602079729

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.602079729 and -0.602079729.

Subproblem 1
x + 0.25 = 0.602079729

Simplifying
x + 0.25 = 0.602079729

Reorder the terms:
0.25 + x = 0.602079729

Solving
0.25 + x = 0.602079729

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = 0.602079729 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 0.602079729 + -0.25
x = 0.602079729 + -0.25

Combine like terms: 0.602079729 + -0.25 = 0.352079729
x = 0.352079729

Simplifying
x = 0.352079729

Subproblem 2
x + 0.25 = -0.602079729

Simplifying
x + 0.25 = -0.602079729

Reorder the terms:
0.25 + x = -0.602079729

Solving
0.25 + x = -0.602079729

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = -0.602079729 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -0.602079729 + -0.25
x = -0.602079729 + -0.25

Combine like terms: -0.602079729 + -0.25 = -0.852079729
x = -0.852079729

Simplifying
x = -0.852079729

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.352079729, -0.852079729}
Solution
x = {0.352079729, -0.852079729}