6x^2-9x-54=0

Solution for 6x^2-9x-54=0 equation:

Simplifying
6x2 + -9x + -54 = 0

Reorder the terms:
-54 + -9x + 6x2 = 0

Solving
-54 + -9x + 6x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '3'.
3(-18 + -3x + 2x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-18 + -3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-18 + -3x + 2x2 = 0

Solving
-18 + -3x + 2x2 = 0

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

Divide each side by '2'.
-9 + -1.5x + x2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + -1.5x + 9 + x2 = 0 + 9

Reorder the terms:
-9 + 9 + -1.5x + x2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + -1.5x + x2 = 0 + 9
-1.5x + x2 = 0 + 9

Combine like terms: 0 + 9 = 9
-1.5x + x2 = 9

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 9 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 9 + 0.5625

Combine like terms: 9 + 0.5625 = 9.5625
0.5625 + -1.5x + x2 = 9.5625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(x + -0.75) equal to 3.092329219 and -3.092329219.

Subproblem 1
x + -0.75 = 3.092329219

Simplifying
x + -0.75 = 3.092329219

Reorder the terms:
-0.75 + x = 3.092329219

Solving
-0.75 + x = 3.092329219

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = 3.092329219 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = 3.092329219 + 0.75
x = 3.092329219 + 0.75

Combine like terms: 3.092329219 + 0.75 = 3.842329219
x = 3.842329219

Simplifying
x = 3.842329219

Subproblem 2
x + -0.75 = -3.092329219

Simplifying
x + -0.75 = -3.092329219

Reorder the terms:
-0.75 + x = -3.092329219

Solving
-0.75 + x = -3.092329219

Solving for variable 'x'.

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

Add '0.75' to each side of the equation.
-0.75 + 0.75 + x = -3.092329219 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + x = -3.092329219 + 0.75
x = -3.092329219 + 0.75

Combine like terms: -3.092329219 + 0.75 = -2.342329219
x = -2.342329219

Simplifying
x = -2.342329219

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.842329219, -2.342329219}
Solution
x = {3.842329219, -2.342329219}