1.5x^2+15x+31.5=0

Solution for 1.5x^2+15x+31.5=0 equation:

Simplifying
1.5x2 + 15x + 31.5 = 0

Reorder the terms:
31.5 + 15x + 1.5x2 = 0

Solving
31.5 + 15x + 1.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '1.5'.
21 + 10x + x2 = 0

Move the constant term to the right:

Add '-21' to each side of the equation.
21 + 10x + -21 + x2 = 0 + -21

Reorder the terms:
21 + -21 + 10x + x2 = 0 + -21

Combine like terms: 21 + -21 = 0
0 + 10x + x2 = 0 + -21
10x + x2 = 0 + -21

Combine like terms: 0 + -21 = -21
10x + x2 = -21

The x term is 10x.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10x + 25 + x2 = -21 + 25

Reorder the terms:
25 + 10x + x2 = -21 + 25

Combine like terms: -21 + 25 = 4
25 + 10x + x2 = 4

Factor a perfect square on the left side:
(x + 5)(x + 5) = 4

Calculate the square root of the right side: 2

Break this problem into two subproblems by setting 
(x + 5) equal to 2 and -2.

Subproblem 1
x + 5 = 2

Simplifying
x + 5 = 2

Reorder the terms:
5 + x = 2

Solving
5 + x = 2

Solving for variable 'x'.

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

Add '-5' to each side of the equation.
5 + -5 + x = 2 + -5

Combine like terms: 5 + -5 = 0
0 + x = 2 + -5
x = 2 + -5

Combine like terms: 2 + -5 = -3
x = -3

Simplifying
x = -3

Subproblem 2
x + 5 = -2

Simplifying
x + 5 = -2

Reorder the terms:
5 + x = -2

Solving
5 + x = -2

Solving for variable 'x'.

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

Add '-5' to each side of the equation.
5 + -5 + x = -2 + -5

Combine like terms: 5 + -5 = 0
0 + x = -2 + -5
x = -2 + -5

Combine like terms: -2 + -5 = -7
x = -7

Simplifying
x = -7

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3, -7}