x(1.5x+3)=120

Solution for x(1.5x+3)=120 equation:

Simplifying
x(1.5x + 3) = 120

Reorder the terms:
x(3 + 1.5x) = 120
(3 * x + 1.5x * x) = 120
(3x + 1.5x2) = 120

Solving
3x + 1.5x2 = 120

Solving for variable 'x'.

Reorder the terms:
-120 + 3x + 1.5x2 = 120 + -120

Combine like terms: 120 + -120 = 0
-120 + 3x + 1.5x2 = 0

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

Divide each side by '1.5'.
-80 + 2x + x2 = 0

Move the constant term to the right:

Add '80' to each side of the equation.
-80 + 2x + 80 + x2 = 0 + 80

Reorder the terms:
-80 + 80 + 2x + x2 = 0 + 80

Combine like terms: -80 + 80 = 0
0 + 2x + x2 = 0 + 80
2x + x2 = 0 + 80

Combine like terms: 0 + 80 = 80
2x + x2 = 80

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 80 + 1

Reorder the terms:
1 + 2x + x2 = 80 + 1

Combine like terms: 80 + 1 = 81
1 + 2x + x2 = 81

Factor a perfect square on the left side:
(x + 1)(x + 1) = 81

Calculate the square root of the right side: 9

Break this problem into two subproblems by setting 
(x + 1) equal to 9 and -9.

Subproblem 1
x + 1 = 9

Simplifying
x + 1 = 9

Reorder the terms:
1 + x = 9

Solving
1 + x = 9

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 9 + -1

Combine like terms: 1 + -1 = 0
0 + x = 9 + -1
x = 9 + -1

Combine like terms: 9 + -1 = 8
x = 8

Simplifying
x = 8

Subproblem 2
x + 1 = -9

Simplifying
x + 1 = -9

Reorder the terms:
1 + x = -9

Solving
1 + x = -9

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -9 + -1

Combine like terms: 1 + -1 = 0
0 + x = -9 + -1
x = -9 + -1

Combine like terms: -9 + -1 = -10
x = -10

Simplifying
x = -10

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8, -10}