1.5x^2+1.5x-84=0

Solution for 1.5x^2+1.5x-84=0 equation:

Simplifying
1.5x2 + 1.5x + -84 = 0

Reorder the terms:
-84 + 1.5x + 1.5x2 = 0

Solving
-84 + 1.5x + 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'.
-56 + x + x2 = 0

Move the constant term to the right:

Add '56' to each side of the equation.
-56 + x + 56 + x2 = 0 + 56

Reorder the terms:
-56 + 56 + x + x2 = 0 + 56

Combine like terms: -56 + 56 = 0
0 + x + x2 = 0 + 56
x + x2 = 0 + 56

Combine like terms: 0 + 56 = 56
x + x2 = 56

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

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 56 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 56 + 0.25

Combine like terms: 56 + 0.25 = 56.25
1.25 + x2 = 56.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 56.25

Calculate the square root of the right side: 7.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 7.5 and -7.5.

Subproblem 1
x + 0.5 = 7.5

Simplifying
x + 0.5 = 7.5

Reorder the terms:
0.5 + x = 7.5

Solving
0.5 + x = 7.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 7.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 7.5 + -0.5
x = 7.5 + -0.5

Combine like terms: 7.5 + -0.5 = 7
x = 7

Simplifying
x = 7

Subproblem 2
x + 0.5 = -7.5

Simplifying
x + 0.5 = -7.5

Reorder the terms:
0.5 + x = -7.5

Solving
0.5 + x = -7.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -7.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -7.5 + -0.5
x = -7.5 + -0.5

Combine like terms: -7.5 + -0.5 = -8
x = -8

Simplifying
x = -8

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7, -8}