3x(2+3x)=45

Solution for 3x(2+3x)=45 equation:

Simplifying
3x(2 + 3x) = 45
(2 * 3x + 3x * 3x) = 45
(6x + 9x2) = 45

Solving
6x + 9x2 = 45

Solving for variable 'x'.

Reorder the terms:
-45 + 6x + 9x2 = 45 + -45

Combine like terms: 45 + -45 = 0
-45 + 6x + 9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-15 + 2x + 3x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-15 + 2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-15 + 2x + 3x2 = 0

Solving
-15 + 2x + 3x2 = 0

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

Divide each side by '3'.
-5 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 0.6666666667x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + 0.6666666667x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 0.6666666667x + x2 = 0 + 5
0.6666666667x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
0.6666666667x + x2 = 5

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 5 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 5 + 0.1111111112

Combine like terms: 5 + 0.1111111112 = 5.1111111112
0.1111111112 + 0.6666666667x + x2 = 5.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 5.1111111112

Calculate the square root of the right side: 2.260776661

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 2.260776661 and -2.260776661.

Subproblem 1
x + 0.3333333334 = 2.260776661

Simplifying
x + 0.3333333334 = 2.260776661

Reorder the terms:
0.3333333334 + x = 2.260776661

Solving
0.3333333334 + x = 2.260776661

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 2.260776661 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 2.260776661 + -0.3333333334
x = 2.260776661 + -0.3333333334

Combine like terms: 2.260776661 + -0.3333333334 = 1.9274433276
x = 1.9274433276

Simplifying
x = 1.9274433276

Subproblem 2
x + 0.3333333334 = -2.260776661

Simplifying
x + 0.3333333334 = -2.260776661

Reorder the terms:
0.3333333334 + x = -2.260776661

Solving
0.3333333334 + x = -2.260776661

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -2.260776661 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -2.260776661 + -0.3333333334
x = -2.260776661 + -0.3333333334

Combine like terms: -2.260776661 + -0.3333333334 = -2.5941099944
x = -2.5941099944

Simplifying
x = -2.5941099944

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.9274433276, -2.5941099944}
Solution
x = {1.9274433276, -2.5941099944}