4x^2+48x+54=0

Solution for 4x^2+48x+54=0 equation:

Simplifying
4x2 + 48x + 54 = 0

Reorder the terms:
54 + 48x + 4x2 = 0

Solving
54 + 48x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(27 + 24x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(27 + 24x + 2x2)' equal to zero and attempt to solve:

Simplifying
27 + 24x + 2x2 = 0

Solving
27 + 24x + 2x2 = 0

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

Divide each side by '2'.
13.5 + 12x + x2 = 0

Move the constant term to the right:

Add '-13.5' to each side of the equation.
13.5 + 12x + -13.5 + x2 = 0 + -13.5

Reorder the terms:
13.5 + -13.5 + 12x + x2 = 0 + -13.5

Combine like terms: 13.5 + -13.5 = 0.0
0.0 + 12x + x2 = 0 + -13.5
12x + x2 = 0 + -13.5

Combine like terms: 0 + -13.5 = -13.5
12x + x2 = -13.5

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = -13.5 + 36

Reorder the terms:
36 + 12x + x2 = -13.5 + 36

Combine like terms: -13.5 + 36 = 22.5
36 + 12x + x2 = 22.5

Factor a perfect square on the left side:
(x + 6)(x + 6) = 22.5

Calculate the square root of the right side: 4.74341649

Break this problem into two subproblems by setting 
(x + 6) equal to 4.74341649 and -4.74341649.

Subproblem 1
x + 6 = 4.74341649

Simplifying
x + 6 = 4.74341649

Reorder the terms:
6 + x = 4.74341649

Solving
6 + x = 4.74341649

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = 4.74341649 + -6

Combine like terms: 6 + -6 = 0
0 + x = 4.74341649 + -6
x = 4.74341649 + -6

Combine like terms: 4.74341649 + -6 = -1.25658351
x = -1.25658351

Simplifying
x = -1.25658351

Subproblem 2
x + 6 = -4.74341649

Simplifying
x + 6 = -4.74341649

Reorder the terms:
6 + x = -4.74341649

Solving
6 + x = -4.74341649

Solving for variable 'x'.

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

Add '-6' to each side of the equation.
6 + -6 + x = -4.74341649 + -6

Combine like terms: 6 + -6 = 0
0 + x = -4.74341649 + -6
x = -4.74341649 + -6

Combine like terms: -4.74341649 + -6 = -10.74341649
x = -10.74341649

Simplifying
x = -10.74341649

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.25658351, -10.74341649}
Solution
x = {-1.25658351, -10.74341649}