3x*3x+6x=180

Solution for 3x*3x+6x=180 equation:

Simplifying
3x * 3x + 6x = 180

Reorder the terms for easier multiplication:
3 * 3x * x + 6x = 180

Multiply 3 * 3
9x * x + 6x = 180

Multiply x * x
9x2 + 6x = 180

Reorder the terms:
6x + 9x2 = 180

Solving
6x + 9x2 = 180

Solving for variable 'x'.

Reorder the terms:
-180 + 6x + 9x2 = 180 + -180

Combine like terms: 180 + -180 = 0
-180 + 6x + 9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-60 + 2x + 3x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-60 + 2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-60 + 2x + 3x2 = 0

Solving
-60 + 2x + 3x2 = 0

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

Divide each side by '3'.
-20 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 0.6666666667x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 0.6666666667x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 0.6666666667x + x2 = 0 + 20
0.6666666667x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
0.6666666667x + x2 = 20

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 = 20 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 20 + 0.1111111112

Combine like terms: 20 + 0.1111111112 = 20.1111111112
0.1111111112 + 0.6666666667x + x2 = 20.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 20.1111111112

Calculate the square root of the right side: 4.484541349

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 4.484541349 and -4.484541349.

Subproblem 1
x + 0.3333333334 = 4.484541349

Simplifying
x + 0.3333333334 = 4.484541349

Reorder the terms:
0.3333333334 + x = 4.484541349

Solving
0.3333333334 + x = 4.484541349

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 = 4.484541349 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 4.484541349 + -0.3333333334
x = 4.484541349 + -0.3333333334

Combine like terms: 4.484541349 + -0.3333333334 = 4.1512080156
x = 4.1512080156

Simplifying
x = 4.1512080156

Subproblem 2
x + 0.3333333334 = -4.484541349

Simplifying
x + 0.3333333334 = -4.484541349

Reorder the terms:
0.3333333334 + x = -4.484541349

Solving
0.3333333334 + x = -4.484541349

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 = -4.484541349 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -4.484541349 + -0.3333333334
x = -4.484541349 + -0.3333333334

Combine like terms: -4.484541349 + -0.3333333334 = -4.8178746824
x = -4.8178746824

Simplifying
x = -4.8178746824

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.1512080156, -4.8178746824}
Solution
x = {4.1512080156, -4.8178746824}